General Identities
Notations
Traditional name
Abstract sufficiently smooth (analytic or piecewise differentiable) function
Traditional notation
f z
Mathematica StandardForm notation H L f z
Primary@ D definition
f z
The arbitrary function f z in this document is defined for all complex z or for some region of c (if it is described explicitly).H L In the majority of cases, it is assumed to be an analytical function of the variable z (and sufficiently fast decaying if needed for the convergence of integrals and sums). H L
General characteristics
Domain and analyticity
In the majority of cases, f z is an analytical function of z, which is defined in the whole complex z plane (if special restrictions are not shown).
z f z CC H L
Symmetries and periodicities H L f z + f -z f z - f -z f -z f z ; fo z 0 f z fe z + fo z fe z fo z 2 2
This formula is the condition for a function to be even. HForL example,H L the functionH L f Hz L cos z is an even function. H L H L H L í H L H L H L í H L í H L f z + f -z f z - f -z f -z - f z ; fe z 0 f z fe z + fo z fe z fo z 2 2H L H L
This formula is the condition for a function to be odd. ForH Lexample,H L the functionH Lf z H sinL z is an odd function. H L H L H L í H L H L H L í H L í H L
H L H L http://functions.wolfram.com 2
f z + m Ρ f z ; m Î Z
This formula reflects periodicity of function f z (if f is periodic with period Ρ). The analytic function f z is calledH periodicL H Lif there exists a complex constant Ρ ¹ 0 , such that f z + Ρ f z ; z Î C. The number Ρ (with minimal H L H L value Ρ ) is called the period of the function f z . For example, the functions f z sin z and f z tan z have periods Ρ 2 Π and Ρ Π accordingly. H L H L Ρ 2 Π ¤ H L H L H L H L H L
Series representations
General remarks
There are three main possibilities to represent an arbitrary function f z as an infinite sum of simple functions. The first is the power series expansion and its two important generalizations, the Laurent series and the Puiseux series. The second is the q-series and Dirichlet series (general and periodic), and the third is the Fourier series H L (exponential, trigonometric, and generalized Fourier series by the orthogonal systems). These representations provide very general convenient methods for studying a wide range of functions.
k The terms of a power series expansion or its generalizations include power functions in the form z - z0 or k m k z - z0 ; the terms in the q-series include expressions like q ; q Φ z ; the terms in a Dirichlet series include
exponential functions in the form exp -Λk s . The Fourier trigonometric series usually provide expansions in terms H L of cos k x and sin k x (for trigonometric series) and in terms of ãä k x (for exponential series). Generalized Fourier H L H L series provide expansions in terms of other orthogonal systems of functions, such as the classical orthogonal H L polynomials. With each of these methods, you can express in closed form an enormous number of non-elementary functionsH Lin terms Hof onlyL simple elementary functions.
Generalized power series
Expansions at z z0
For the function itself
¥ f k z 0 k f z « z - z0 ; z - z0 < R £ ¥ k! k=0 H L H L TheH L Taylorâ seriesH, firstL investigated ¤ by B. Taylor in 1715, gives the above Taylor expansion for an arbitrary function f z at a finite point z z0.
H L http://functions.wolfram.com 3
The sign « means that the Taylor series should not be taken as purely equal to f z , since it may not converge
everywhere on the complex plane. The series converges absolutely in some disk of radius R centered on z0, where R is called the radius of convergence. On the disk z - z < R, you can exchange the sign « for . You can state 0 H L f k z for most functions that R = lim c -1 k ; c = 0 . The Taylor series coefficients c can also be evaluated by k®¥ k k k! k H L 1 f t ¤ the Cauchy integral formula ck = ât ;ICMΡ t - z0 Ρ < R. 2 Π ä CΡ k+1 t-z0 ¤ H L There are then three cases for R: either R is infinite, R is zero, or R is some finite positive number greater than 0. Ù I M ¤
If R = 0, this series converges only at the point z0, and the Taylor series offers little analytical benefit. For example, ¥ k the series k=0 k! z - z0 has no disk of convergence around z0, since the coefficient k! makes the terms grow
infinitely large no matter how small z - z0 becomes.
If R = ¥, Úthe seriesH convergesL in the entire plane. In such cases it either represents an entire transcendental func- ¥ ¤ k tion—as, for example, the series k=0 z - z0 k! does for the exponential function exp z - z0 —or it contains only a finite number of terms and therefore represents a polynomial.
If 0 < R < ¥, the sum of the seriesÚ definesH aL regular analytic function having at least oneH singularL point on the
circle z - z0 R. There may be finitely or infinitely many singular points on the circle, but there must be at least ¥ k one. The power series k=1 z - z0 k = -log z0 + 1 - z , for example, has only one singular point, at z = z0 + 1. Other power series have dense sets of singular points on the circle, such as the series ¥ zk!, which has many ¤ k=0 singular points on the unit circle, the edge of its natural region of analyticity. The same holds true, with the same Ú H L H L region of analyticity, for the series representation of the logarithm of InverseEllipticNomeQ: Ú q-1 z 2 k 8 log z = ¥ log 1+z . 16 k=1 1+z2 k+1
H L Inside the region of convergence of the series, you can exchange the sign « for in the preceding Taylor expan- J N Ú J N sion for an arbitrary function f z at a finite point z0:
¥ f k z 0 k f z z - z0 ; z - z0 < R £ ¥ k! H L k=0 H L H L InfiniteH L â Taylor seriesH expansionL ¤can be approximated by the truncated version, the finite Taylor polynomial expan- sions. But in these cases, instead of the equality sign , you will use the sign µ and an ellipsis … or an explicit Landau O term, for example:
1 1 ¢ ¢¢ 2 3 3 f z µ f z0 + f z0 z - z0 + f z0 z - z0 + f z0 z - z0 + ¼ ; z ® z0 2 6 H L n f k z H L H L 0H L H k L H LnH L H L H L H L f z µ z - z0 + O z - z0 k! k=0 H L
H L n TheH L Landauâ O termH OL z - HzH0 inL Lthe preceding relation means that the next expression is bounded near point
z z0:
HH L L http://functions.wolfram.com 4
n k k f z0 z - z0 -n z - z0 f z - < const k! k=0 H L H L H L For compositionsH L H L withâ elementary functions
Including power functions
Π - arg c - Im m log z - z0 a f z r b exp 2 Π ä b r + 2 Π
H L H H LL z H H L L r Π - arg a cr - arg Π - arg a c - Im m r log z - z0 Π - arg c - Im m log z - z0 a cr - r + 2 Π 2 Π 2 Π
H L H H LL H L H H LL H L J N ¥ k r k -1 -b k f z f z a cr b z - z m r b c-r - 1 ; z ® z lim c c ¹ 0 0 m 0 m z®z0 k=0 k! z - z0 z - z0 H L H L H L H L H L H L â H L í í Including logarithmic and power functions H L H L
Π - arg c - Im m log z - z0 log a f z r 2 ä Π r + 2 Π
H L H H LL z H H L L r Π - arg a cr - arg Π - arg a c - Im m r log z - z0 Π - arg c - Im m log z - z0 a cr - r + + 2 Π 2 Π 2 Π
H L H H LL H L H H LL H L J N ¥ -1 k-1 f z r k f z log a cr + m r log z - z + c-r - 1 ; z ® z lim c c ¹ 0 0 m 0 m z®z0 k=1 k z - z0 z - z0 H L H L H L Π - Im log q f z Π - arg a - Im r log q f z Π - Im log q f z f Hz r L H L â 0 H 0L í í 0 log a q 2 ä Π r + H L - Re r H L + 2 Π 2 Π 2 Π
H L ¥ k k f z0H zH-Lz0 H LL H L H H L H LL H H L H LL I Ilog aM M+ r log q ; z ® z0 H L k! k=0 H L H L H L H L H L â H L StringTake[" |||", {6, 4}] Expansions at z 0
For the function itself
1 1 ¢ ¢¢ 2 3 3 f z µ f 0 + f 0 z + f 0 z + f 0 z + ¼ ; z ® z0 2 6 H L 1 1 f HzL µ f H0L + f ¢H0L z + f ¢¢H0L z2 + f 3 H0L z3 + O z4 H L 2 6 H L H L H L H L H L H L I M http://functions.wolfram.com 5
¥ f k 0 f z « zk ; z < R £ ¥ k! k=0 H L H L f HzL µ fâ0 1 + O z ¤
Laurent series H L H L H H LL ¥ 1 f t k f z « ck z - z0 ; ck ât k Î Z C k+1 k=-¥ 2 Π ä Ρ t - z0 H L TheH L Laurentâ seriesH ,L first studied àby P. Laurent iní 1843, gives the Laurent expansion for the function f z around a H L finite point z0, where CΡ is the circle t - z0 = Ρ with radius Ρ, r < Ρ < R. Here R is the radius of convergence of its regular part ¥ c z - z k and 1 r is the radius of convergence of its principal part k=0 k 0 H L -1 k ¥ -k -1 k=-¥ ck z - z0 = k=1 c-k z - z0 . The coefficient¤ c-1 of the power z - z0 in the Laurent expansion of function f z is calledÚ the HresidueL of f z at the point z0: Ú H L 1 Ú H L H L resz f z z0 f t ât H L 2 Π ä CΡ H L
If r < R, then the Laurent series converges absolutely in the ring r < z - z < R, where its sum defines an analytic H H LL H L à H L 0 function. If r = 0 and the principal part includes only a finite number of terms (that is, if ck = 0 for k < -n < 0 for + some n Î N ), then f z has a pole of order n at the point z = z0. ¤ For example, the function cot z has a pole of order 1 (also called a simple pole) at the point z = 0, since it has only one term in the principalH L part of its Laurent series expansion:
¥ k 2 k 3 1 -1 2 B2 k H L 1 z z cot z + z2 k-1 µ - - + º z k=1 2 k ! z 3 45 H L If rH =L 0 and âthe principal part includes an infinite number of terms, this analytic function has an essential singular- H L 0 k ity at the point z = z0. For example, the sum k=-¥ z -k ! = exp 1 z has an essential singularity at the point z = 0.
Ú H L H L Puiseux series
¥ k m f z « ck z - z0 k=-¥ TheH L Puiseuxâ HseriesL, first studied by V.-A. Puiseux in 1850, expand f z around a finite point z0 as ¥ k m f z « k=-¥ ck jk z - z0 , with jk w usually chosen as jk w = w . (Other choices for jk w are possible k m r k m r s through the use of iterated logarithms of the form jk w = w log w , jk w w log w log log w , and so H L k m on.) Choosing jk w = w gives the Puiseux series for algebraic bivariate functions (because such functions H L Ú H L H L H L H L should not include logarithms like log r w ). H L H L H L H L H H LL H L H L http://functions.wolfram.com 6
If this series contains only a finite number of nonzero coefficients ck with negative indices k, the point z0 is an algebraic branch point of order m - 1; otherwise it is a transcendental branch point, such as the point z = 0 for the function exp 1 z .
Puiseux series are widely used for describing solutions of differential equations near singular points and, in particu- lar, for representationsH L of elementary and special functions near their singular points. The named elementary functions can have rather complicated behaviors near their singular points.
k 1 k ¥ -1 z - 1 2 k cos-1 z 2 1 - z ; z - 1 < 1 k k=0 2 2 k + 1 k! H L J N H L The H pointL z = 1 isâ an algebraic branch ¤ point of order 1 for the function H L cos-1 z 1 - z z - 1 log z + z - 1 z + 1 . This function has the Puiseux representation for its fundamental branch near point z = 1.
A similarH L representationI occursN nearI the branch point z N= -1.
1 -2 k ¥ z Π z 1 2 k cos-1 z - log -4 z2 - ; z > 1 2 2 k k! 2 -z2 k=1 J N This HexpansionL near the branchI pointM âz = ¥ shows that ¤ cos-1 z has a logarithmic-type singularity that can provide infinitely many values depending on the direction of approach of the variable z to ¥.
¥ zk logk z H L zz k=0 k!
H L z This âPuiseux series for z near the point z = 0 includes powers of logarithmic functions.
The Puiseux series for log2 log z near the point z = 0 coincides with itself.
-j+k+1 ¥ k k j -1 Sk log log z W z log z - log log z - H H LL µ k k=0 log z j=1 H L j!
H L 2 H H LL H L H L H H LlogL âlog z logâ log z - log log z log z - log log z + + + º ; z ® ¥ H L 2 log z 2 log z H H LL H H LL H H LL This PuiseuxH L seriesH H L Lfor the ProductLog function W 1 z near theH ¤ pointL z = ¥ has a more complicated structure involving iterated logarithms.H L The complicatedH Lstructure is to be expected from inverse functions like ProductLog. H L q-series
¥ k f z « ck q ; q Φ z k=-¥
H L â H L http://functions.wolfram.com 7
Π This q-series is widely used in applications. In the case q Φ z ãä z, c 1 f t ã-ä k t ât, it coincides with k 2 Π -Π exponential Fourier series.
In very particular cases, q-series can coincide with some H trigonometricL functions.Ù H L For example, if q Φ z ãä z, c c 1 , and all other c are zero, then f z cos z . -1 1 2 k
In more complicated cases, q-series can define different special functions, for example elliptic theta functions: H L H L H L ¥ 2 cos 2 k w k Î Z k > 0 k J3 w, q 1 + q . k=-¥ 0 True J N í DirichletH L seriesâ
General Dirichlet series
¥ log ak f z « ak exp -Λk z ; Re z > c c = lim k®¥ k=0 Λk H ¤L IfH allL coefficientsâ H Λk >L 0, thisH L Dirichletß series for the function f z converges in some open half-plane Re z > c. In this case, for many Dirichlet series the abscissa of absolute convergence can be found by the formula
log ak c limk®¥ . Λk H L H L
I¡ ¥M In the more general case 0 < Λ0 £ Λ1 £ ¼, the Dirichlet series absolutely converges in some open convex domain. In both cases, the sum of the Dirichlet series is an analytic function in the domain of convergence.
k For example, if Λk = 2 k and ak = ¤1 or ak¤ = -1 , you have, respectively, the Dirichlet series:
¥ csch z ã-2 k z ã z ; Re z > 0 H L k=0 2
¥ H zL â ã sech zH L -1 k ã-2 k z - ; Re z > 0 k=0 2
H L ¥ z âIn theH Lcase Λk = log k , you haveH L the ordinary Dirichlet series f z « k=1 ak k , which in its simplest case ¥ z (ak = 1, Re z > 1) is the Riemann zeta function Ζ z = k=1 1 k . Other interesting examples of the Dirichlet 2 series include the series for elliptic theta functions. For instance, if a = 1 qk and Λ = ¡2kä, you have the represen- H L H Lk Ú2 H kL tations: H L H L Ú H L
¥ ¥ ¥ 2 1 2 1 2 qk cos 2 k z qk ã2 k ä z + qk ã-2 k ä z ; q < 1 k=1 2 k=1 2 k=1
¥ 1 â k2 H L â â ¤ q cos 2 k z J3 z, q - 1 ; q < 1 k=1 2
Generalizedâ H LFourierH H seriesL L ¤ http://functions.wolfram.com 8
¥ b b f x « dk Ψk x ; dk Ψk t f t ât k Î N Ψm t Ψn t ât ∆m,n k=0 a a
UnderH L someâ additionalH L conditionsà H L H (suchL í as piecewiseíà differentiability),H L H L this Fourier series of an arbitrary function f x by the orthogonal system Ψk x with Fourier coefficients dk converges to f x on an interval a, b at the points of continuity of f , and to 1 f x + 0 + f x - 0 at the points of discontinuity of f , where 2 f x ± 0 = lim f x ± Ε ). H L Ε®+0 8 H L< H L H L H H L H LL ¥ b b b 2 2 H dk £L f t âHt ; dk L Ψk t f t ât k Î N Ψm t Ψn t ât ∆m,n k=0 a a a
âThe inequalityà H L takes placeà for allH L orthogonalH L í systemsíà Ψk HxL ; itH Lis called Bessel's inequality. If it can be transformed into Parseval's equality:
¥ b b b 2 2 8 H L< dk f t ât ; dk Ψk t f t ât k Î N Ψm t Ψn t ât ∆m,n k=0 a a a
âThe correspondingà H L systemà Ψk HxL isH Lcalledí the completeíà system.H L H L
The best-known examples of orthogonal systems are the classical orthogonal polynomials and the trigonometric system including cos k x and8 sinH L Generalized Fourier series through classical orthogonal polynomials H L H L ¥ b b f x « dk Ψk x ; dk Ψk t f t ât k Î N Ψm t Ψn t ât ∆m,n k=0 a a ForH L example,â HanyL sufficientlyà H L smoothH L í functioní fàx canH L beH Lexpanded in the Hermite orthogonal system Hn x n=0,1,¼ or in other orthogonal systems with corresponding weight factors as a generalized Fourier series, with its sum converging to f x almost everywhere in corresponding intervals of variable x. The following repre- H L sents this type of Fourier expansion through all classical orthogonal systems of polynomials. 8 H L< ¥ ¥ x2 H L 1 - f x dn Ψn x ; dn Ψn t f t ât Ψn x ã 2 Hn x x Î R -¥ n=0 Π 2n n! H L ¥ H L ¥ H L H L í H L x H L í â à - f x dn Ψn x ; dn Ψn t f t ât Ψn x ã 2 Ln x x > 0 n=0 0 ¥ H L â H L ¥ H L H L í H L HnL! í x à Λ 2 - Λ f x dn Ψn x ; dn Ψn t f t ât Ψn x x ã 2 Ln x x > 0 n=0 0 G n + Λ + 1 H L ⥠H L à 1 H L H L í H L 1 H L í f x dn Ψn x ; dn Ψn t f t ât Ψn x H 2 n + 1 LPn x -1 < x < 1 n=0 -1 2 H L â H L à H L H L í H L H L H L í http://functions.wolfram.com 9 2 ¥ 1 Π 1 - 2 4 f x dn Ψn x ; dn Ψn t f t ât Ψn x 1 - x Tn x -1 < x < 1 -1 n=0 2 - 1 ∆n + 1 H L ⥠H L à 1 H L H L í H L 2 I M H L í 4 2 f x dn Ψn x ; dn Ψn t f t ât Ψn x I 1N - x Un x -1 < x < 1 n=0 -1 Π 1 Λ- H L ⥠H L à 1 H L H L í H L 2 2 n! n + ΛH GL íΛ 2 Λ-1 2 4 Λ f x dn Ψn x ; dn Ψn t f t ât Ψn x 1 - x Cn x -1 < x < 1 n=0 -1 Π G n + 2 Λ ¥ H L H L â H L 1 H L H L í H L I M H L í f x d Ψ x ; d à Ψ t f t ât n n n n H L n=0 -1 a+b+1 - 2 2 n! a + b + 2 n + 1 G a + b + n + 1 H L â H L à H L H L í a 2 b 2 a,b Ψn x 1 - x x + 1 Pn x -1 < x < 1 G a + n + 1 G b + n + 1 H L H L ExponentialH L Fourier series H L H L H L í H L H L ¥ 1 Π ä k x -ä k t f x « ck ã ; ck f t ã ât k=-¥ 2 Π -Π ThisH L exponentialâ Fourier seriesà expansionH L can be transformed into a trigonometric Fourier series by using the Euler formula ãä z cos z + ä sin z . Following is an example of the exponential Fourier series for the simple function f x x ; -Π < x < Π. H L H L ¥ 0 k 0 ä k x x c ã ; c ä k Π cos k Π -sin k Π -Π < x < Π k k True H L k=-¥ k2 Π H H L H LL Outsideâ the interval -Π < x < Π, the sum formsí a periodic function and the following expansion holds for all real x: x + Π -Π x-Π Î Z ¥ 0 k 0 2 Π ä k x x 2 Π + + c ã ; c ä k Π cos k Π -sin k Π x Î R k k True 2 Π 0 True k=-¥ k2 Π H H L H LL Trigonometric Fourier seriesâ í ¥ a0 1 Π 1 Π f x « + ak cos k x + bk sin k x ; ak f t cos k t ât bk f t sin k t ât 2 k=1 Π -Π Π -Π ThisH L series expansionâH H intoL a trigonometricH LL systemà firstH L appearedH L í in an 1807à paperH L ofH J.L Fourier, who used a similar expansion on the interval 0, 2 Π . However, L. Euler had discovered similar formulas for Fourier coefficients in 1777. H L http://functions.wolfram.com 10 An arbitrary interval a, b can be transformed into the interval -Π, Π by changing the variable x ® x b - a 2 Π + a + b 2. This allows you to write the following Fourier series expansion of the arbitrary function f x on an arbitrary interval a, b : H L H L ¥ H a0 L H L H L a + b 2 Π a + b 2 Π f x « + a cos k x - + b sin k x - ; H L k H L k 2 k=1 2 b - a 2 b - a 2 b a + b Π 2 b a + b Π HakL â f t cos k t - 2 ât bk f t sin k t - 2 ât. b - a a 2 b - a b - a a 2 b - a In the preceding formulas the coefficients should vanish at infinity ak ® 0, bk ® 0 as k ® ¥. à H L í à H L In the internal points x, a < x < b, where the piecewise differentiable function f x is continuous, the preceding sum of the Fourier series is equal to f x and you can change the symbol « to . At the points of discontinuity, this Fourier sum is equal to 1 f x + 0 + f x - 0 . H L 2 H L Outside the interval a, b , the Fourier sum is a periodic function with period b - a, but the behavior of this sum is not necessarily related to the actual behavior of the function Hf Hx outsideL theH intervalLL of expansion a, b . H L sin k Π 2 cos k Π For example, let f x = x and a, b = -1, 1 . Then ak = 0, bk = 2 - , k Î N, and: k2 Π2 k Π H L H L H L H L ¥ 2 sin k Π 2 cos k Π ä f x « - sin k Π x = log ã-Π ä x 2 2 H L H L H L k=1 k Π k Π Π H L H L InspectionH L shows that the Fourier sumH ä L log ã-Π Iä x hasM period 2 and coincides with x on the interval -1, 1 . But at â Π the endpoint x = 1 the Fourier sum is equal to 0, since you are evaluating ¥ 2 sin k Π - 2 cos k Π sin k Π with k=1 k2 Π2 k Π I M H L sin k Π = 0. This result coincides with 1 lim ä log ã-Π ä x + lim ä log ã-Π ä x = 1H 1L - 1 . H L 2 x®1-0 Π x®1+0 Π 2 Ú J N H L AsymptoticH L series expansions K I M I MO H L Expansions at z ¥ ¥ n -k f z - g z k 0 ck z -k = f z µ g z ck z ; c0 ¹ 0 z ® ¥ < cn+1 ; z > R -n-1 k=0 g z z H L H L Ú ThisH L asymptoticH L â series expansionì H ¤ of theL function f z at ¥ includes the¤ main ¤ term c0 g z ; c0 ¹ 0 and other terms -k H L ¥ -k of the form g z ck z . The corresponding formal asymptotic series k=0 ck z is, by definition, formed in such a n -k f z -g z k=0ck z way that the following inequality holds for all n ÎH LN and sufficiently large z : H L < cn+1 . This g z z-n-1 H L asymptotic series can be a divergent or convergent series. If this seriesÚ converges,H itL coincidesH L Ú with the Taylor power series expansion at infinity. ¤ £ H L § ¤ For example, the confluent hypergeometric function U a, b, z has the following asymptotic series expansion through the divergent series 2F0: H L http://functions.wolfram.com 11 1 -a U a, b, z µ z 2F0 a, a - b + 1; ; - ; z ® ¥ z The relation takes place because the following inequality holds for each n Î N and sufficiently large z : H L H ¤ L -1 k a a-b+1 z-k U a, b, z - z-a n k k k=0 k! < const ; z > R ¤ z-a-n-H1 L H L H L H L Ú The function Ei z has a rather interesting asymptotic ¤ expansion at ¥ through the following divergent series: ãz ¥ k! Ei z µ + Π ä sgn Im z ; z ® ¥ k z k=0 z H L TheH L next exampleâ includesH Ha LconvergentL H ¤ L series in an asymptotic expansion. So, you can use the sign instead of µ: 1 -2 k ¥ z z 2 k sin-1 z log -4 z2 - ; z > 1 k k! 2 -z2 k=1 J N In theH Lparticular case nI 0,M theâ first generic formula ¤ for asymptotic expansion can be rewritten in the following form: 1 f z µ g z c0 + O ; c0 ¹ 0 z ® ¥ z f z - c0 g z < c1 ; z > R z Expansions at z z H L H L 0 ì H ¤ L H H H L H LL¤ ¤ L ¥ n k f z - g z ck z - z0 k k=0 f z µ g z ck z - z0 ; c0 ¹ 0 z ® z0 < cn+1 ; z - z0 < Ε n+1 k=0 g z z - z0 H L H L Ú H L ThisH L asymptoticH L â H seriesL expansion ì Hof the Lfunction f z at z z0 includes the main¤ term¤ c0 g z ; c0 ¹ 0 and other k H L H L ¥ k terms g z ck z - z0 . The corresponding formal asymptotic series k=0 ck z - z0 by definition is formed in such a f z -g z n c z-z k H L H Lk=0 k 0 way that the following inequality holds for all n Î N and sufficiently small z - z0 : n+1 < cn+1 . g z z-z0 H L H L Ú H L This asymptotic series can be a divergent or convergent series. If this series converges, HitL coincidesH L Ú I withM the Taylor power series expansion at the point z z0. ¤ H L I M ¤ For example, the functions sec-1 z and sin-1 z have the following asymptotic expansions near the points z 0 and z 1 through convergent series. So, you can use the sign instead of µ here: 1 2 k H L ¥ zH L Π 1 1 4 2 k sec-1 z + z - log - - ; z < 1 2 2 2 2 z z k=1 k k! J N H L 1 âk ¤ ¥ 1 - z Π 2 k sin-1 z - 2 1 - z ; z - 1 < 2 k 2 k=0 2 2 k + 1 k! J N H L H L â ¤ H L http://functions.wolfram.com 12 In the particular case n 0, the first generic formula for asymptotic expansion can be rewritten in the following form: f z - c0 g z f z µ g z c0 + O z - z0 ; c0 ¹ 0 z ® z0 < c1 ; z - z0 < Ε z - z0 H L H L ResidueH H L H LrepresentationsH H LL ì H LL ¤ ¥ f z ress g s a k + b k=0 For manyH L analyticâ H functionsH LL H L f z , you can establish so-called residue representations through infinite sums of residues of another analytic function g z in the points s a k + b. The residue of the analytic function g z in the pole of order m can be calculated by the formula: m-1 m 1 ¶H L g z z - z0 + resz g z z0 lim ; mHÎLN m - 1 ! z®z0 m-1 H L ¶z H H L H L L Φ z If the functionH H LL H gL z can be represented as a quotient g z Η z , where the functions Φ z and Η z are H L analytic at the point z z0 and z0 is the simple root of the equation Η z 0, then the following formula holds: H L H L Φ z H L Φ z H L H L H L res , z, z ; Φ z ¹ 0 Η z 0 Η¢ z ¹ 0 0 ¢ 0 0 0 Η z Η z0 H L Very often,H Lresidue representationsH L appear from the theory of the Meijer G function. The majority of 8 < H L ì H L ì H L functions ofH L the hypergeometricH L type can be equivalently defined as corresponding to infinite sums of s residues from products of ratios of gamma functions G Αk s + Βk on power function z . For example, the following residue representation formula for a logarithm function takes place: 2 ¥ G -s z-s H L log z + 1 ress G s + 1 - j ; z < 1 j=1 G 1 - s H L H L â H L H L ¤ Integral representationsH L Fourier integral representations ¥ 1 ¥ 1 ¥ f x « a t cos t x + b t sin t x ât ; a t f Τ cos t Τ âΤ b t f Τ sin t Τ âΤ 0 Π -¥ Π -¥ The Fourier integral is the continuous analogue of a Fourier series. This formulas can be derived from the Fourier H L à H H L H L H L H LL H L à H L H L í H L à H L H L series expansion of the function f x on interval -l, l as l ® ¥. The substitution of a t and b t into the integral gives the following Fourier integral formulas: H L H L 1 ¥ ¥ f x « f Τ cos t x - Τ âΤ ât Π 0 -¥ H L H L H L à à H L H H LL http://functions.wolfram.com 13 1 ¥ f Τ sin R x - Τ f x « lim âΤ. Π R®¥ -¥ x - Τ The first definitionH canL HbeH rewrittenLL in exponential form, leading to exponential direct and inverse Fourier H L à transforms: ¥ ¥ 1 1 -ä t x ä t Τ f x « FΤ f Τ t ã ât ; FΤ f Τ t f Τ ã âΤ. 2 Π -¥ 2 Π -¥ If HtheL function àf x @is HabsolutelyLD H L integrable @ HonLD theH L real axis,à youH haveL the following equality: R 1 1 -ä t x f x + 0 + f x -H 0L lim FΤ f Τ t ã ât. 2 R®¥ 2 Π -R H H L H LL à @ H LD H L Product representations z log ¥ z 0 ¶log Pk-1 f z k! f z Pk f z ; P0 f z f z Pk f z exp z k=0 ¶z H H H LLL H L ä H H LL H H LL H L í H H LL Transformations Transformations and argument simplifications Argument involving related functions (compositions) ¥ ¥ 1 k k j + f g z ck z ; ck aj b0 pj,k pj,0 1 pj,k j m + m - k bm pj,k-m k Î N k=0 j=0 b0 k m=1 ¥ ¥ H H LL â k â í k í âH L í í f Ζ ak Ζ ; Ζ < r1 g z bk z ; z < r2 b0 ¹ 0 g z < r1 k=0 k=0 This formulaH L â shows how ¤ to generateí H L theâ series expansion ¤ ì of a generalí H compositionL¤ f g z at the point z 0, using the known series expansions for each of the functions f z and g z at z 0. H H LL H L H L http://functions.wolfram.com 14 ¥ k f g z ck z ; k=0 i H cH LL â i + i + + i ; i , i , ¼, i a bi1 bi2 ¼ b k c a c a b c a b + a b2 k 1 2 º k 1 2 k i1+i2+¼+ik 1 2 k 0 0 1 1 1 2 1 2 2 1 i1+2 i2+º+k ik=k â H 3 L 2 í 3 í 4 í í c3 a1 b3 + 2 a2 b1 b2 + a3 b1 c4 a1 b4 + 2 a2 b1 b3 + a2 b2 + 3 a3 b1 b2 + a4 b1 ¼ ¥ ¥ k í k í í f Ζ ak Ζ ; Ζ < r1 g z bk z ; z < r2 b0 0 g z < r1 k=0 k=0 H L â ¤ í H L â ¤ ì í H L¤ This formula shows how to generate series expansion of the general composition f g z at the point z 0, using known series expansions for each function f z and g z at zero. H H LL Products, sums, and powers of the direct function H L H L Products of the direct function ¥ k ¥ ¥ k k k f z g z = an bk-n z ; f z ak z ; z < r1 g z bk z ; z < r2 z < r min r1, r2 k=0 n=0 k=0 k=0 ThisH L formulaH L â showsâ how to multiplyH L â power series ¤ of twoí functionsH L â f z and ¤ g z atí the ¤ point z H0. L n k ¥ n ¥ n k n+1 k k k f z g z = aj bk-j z + z aj+k+1 bn-j z ; f z ak z H ;L z < r H L g z bk z ; z < r k=0 j=0 k=0 j=0 k=0 k=0 ThisH L formulaH L â showsâ how to multiplyâ â power series of twoH functionsL â f z and¤ g íz at theH L pointâ z 0. ¤ ¥ k n k f z g z h z = ak-n ci bn-i z ; H L H L k=0 n=0 i=0 ¥ ¥ ¥ H L H L H L k k k f z âak z â; âz < r1 g z bk z ; z < r2 h z ck z ; z < r3 z < r min r1, r2, r3 k=0 k=0 k=0 This formulaH L â shows how¤ to multiplyí H L powerâ series ¤ of threeí functionsH L â f z , g z¤ and híz at¤ the pointH z 0. L Ratios of the direct function H L H L H L r 1 1 ¥ k -1 k k + 1 p zk ; r r,k g z b0 k=0 r=0 r + 1 ¥ H L 1 k g z âb Hzk Lbâ¹ 0 p 1 p j m + m - k b p k Î N+ H L k 0 j,0 j,k m j,k-m k=0 b0 k m=1 ThisH Lformulaâ representsí theí reciprocalí of a power âseriesH for functionL g z atí the point z 0. H L http://functions.wolfram.com 15 f z ¥ k ¥ ¥ k k k qk z ; ak bj qk-j f z ak z g z bk z g z k=0 j=0 k=0 k=0 H L This formulaâ represents â the ratioí of Ha Lpowerâ seriesí forH Lfunctionsâ f z and g z at the point z 0. H L j f z 1 ¥ k -1 r j j + 1 a p zk ; k-j r r,j H L H L g z b0 k=0 j=0 r=0 r + 1 H L ¥ ¥ H L 1 k ââk H L â k + Hf Lz ak z g z bk z b0 ¹ 0 pj,0 1 pj,k j m + m - k bm pj,k-m k Î N k=0 k=0 b0 k m=1 ThisH Lformulaâ representsí H L theâ ratio íof a powerí series forí functions f zâ andH g z at theL point z í 0. j -1 r f z g z 1 ¥ k j r k H L H L j + 1 dk-j pr,j z ; h z c0 k=0 j=0 r=0 r + 1 H L H L H L ¥ ¥ ¥ âk âH L â k k fH zL ak z ; z < r1 g z bk z ; z < r3 h z ck z c0 ¹ 0 ; z < r2 k=0 k=0 k=0 k 1 k H L â ¤ í H L â ¤ í H L â í ¤ í + z < r min r1, r2, r3 dk an bk-n pj,0 1 pj,k j m - k + m cm pj,k-m k Î N n=0 c0 k m=1 This ¤formula representsH Ltheí ratio âof a powerí series forí functions f zâ, gH z and h zL, at the pointí z 0. Sums of the direct function H L H L H L ¥ ¥ ¥ k k k f z ± g z ak ± bk z ; f z ak z ; z < r1 g z bk z ; z < r2 z < r min r1, r2 k=0 k=0 k=0 ThisH L formulaH L â representsH L the summationH L â property ¤ forí the HpowerL â series of ¤the functionsí ¤ f z andH g zL at the point z 0. Powers of the direct function H L H L ¥ k ¥ 2 k k f z = an ak-n z ; f z ak z ; z < r k=0 n=0 k=0 ThisH L formulaâ â represents the HseriesL â squared property ¤ of the functions f z at the point z 0. ¥ ¥ 1 k f z n p zk ; f z a zk ; z < r p an a ¹ 0 p n j + j - k a p k Î N+ z < r k k 1 0 0 0 k H L j k-j 1 k=0 k=0 a0 k j=1 th ThisH L formulaâ represents H L theâ n integer ¤ powerí of the seriesí for íthe functionsâ f Hz at the pointL z í0. í ¤ H L http://functions.wolfram.com 16 1 arg a0 1 f z 2 ä Α Π - - arg ¥ k -1 j Α 2 2 Π 2 Π a0 Α k - Α k k f z ã a0 Α pj,k z ; J N H L k j k=0 j=0 Α - j ¥ 1 H k L H L k â â + f z ak z a0 ¹ 0 pj,0 1 pj,k j m + m - k am pj,k-m k Î N k=0 a0 k m=1 th ThisH Lformulaâ representsí theí arbitraryí Α power of âtheH series for theL functionsí f z at the point z 0. 1 arg a0 Im Β log z 1 f z 2 ä Α Π - - - arg ¥ k -1 j Β Α 2 2 Π 2 Π 2 Π a0 Α Α Β k - Α k k z f z ã a0 Α z pj,k zH L; J N I H LM H L k j k=0 j=0 Α - j ¥ 1 k H L I H LM k â â + f z ak z a0 ¹ 0 pj,0 1 pj,k j m + m - k am pj,k-m k Î N k=0 a0 k m=1 ThisH Lformulaâ representsí theí generic arbitraryí powerâ ofH the seriesL for the functionsí f z at the point z 0. Related transformations H L Inversion ¥ -1 k f z ck z ; k=0 H L 1 1 k+1 j j j k-1 H L â ip c0 0 c1 ck - aj ¼ ∆ k ∆ k i1 + i2 + ¼ + ik; i1, i2, ¼, ik cp j, ip k, q iq a1 a1 p=1 q=1 j=2 i1=0 i2=0 ik=0 p=1 2 3 Ú Ú 2 í a2 í 2 a2 - a1âa3 â â 5 aâ2 - 5 a1 a3 a2 + a1 a4H Lä í c2 - c3 c4 - ¼ 3 5 7 a1 a1 a1 ¥ ík í -1 í í f Ζ ak Ζ a0 0 a1 ¹ 0 f f z z k=0 H L -1 This formulaH L â representsí the seriesí expansioní I of HtheLM inverse function f z through the known power series of the direct function f z at the point z 0. H L ¥ H L n n n ! a 2 ak k -1 k 1 2 f z ck z ; vk k + n1 + n2 + º + nk - 1; k - 1, n1, n2, ¼, nk - º - H L k a1 a1 k=1 n2+2 n3+º+ k-1 nk=k-1 k a1 H L ¥ H L â k âH L -1H L í f Ζ ak Ζ a0 0 a1 ¹ 0 f f z z k=0 H L -1 This formulaH L â representsí the íseries expansioní I ofH theLM inverse function f z through the known power series of direct function f z at the point z 0. H L Re-expansions in different points H L H L http://functions.wolfram.com 17 ¥ ¥ ¥ j + k j f z f z + a z z - z k ; f z a zk 0 j+k k 0 0 k k=1 j=0 k=0 ThisH L formulaH L âshowsâ how to generateH theL series H L expansionâ of the function f z at the point z z0, if this function is presented through the power series at the point z 0. ¥ ¥ j + k ¥ H L f z a -z j zk ; f z a z - z k j+k k 0 k 0 k=0 j=0 k=0 ThisH L formulaââ shows howH to Lgenerate H Lthe âseriesH expansionL of the function f z at the point z 0, if this function is presented through the power series at the point z z0. ¥ ¥ j + k ¥ H L f z a z - z k z - z j ; f z a z - z k j+k j 1 0 1 k 0 j=0 k=0 k=0 ThisH L formulaââ shows howH to generateL H theL seriesH L expansionâ H ofL the function f z at the point z z1, if this function is presented through the power series at the point z z0. Expansions through classical orthogonal polynomials H L ¥ ¥ ¥ j + k j f z f z + a z z - z k ; f z a zk 0 j+k k 0 0 k k=1 j=0 k=0 ThisH L formulaH L âshowsâ how to generateH theL series H L expansionâ of the function f z at the point z z0, if this function is presented through the power series at the point z 0. ¥ ¥ j + k ¥ H L f z a -z j zk ; f z a z - z k j+k k 0 k 0 k=0 j=0 k=0 ThisH L formulaââ shows howH to Lgenerate H Lthe âseriesH expansionL of the function f z at the point z 0, if this function is presented through the power series at the point z z0. ¥ ¥ j + k ¥ H L f z a z - z k z - z j ; f z a z - z k j+k j 1 0 1 k 0 j=0 k=0 k=0 ThisH L formulaââ shows howH to generateL H theL seriesH L expansionâ H ofL the function f z at the point z z1, if this function is presented through the power series at the point z z0. H L Determinants 2 n-1 1 a1 a1 ¼ a1 2 n-1 n n 1 a2 a2 ¼ a2 ai - aj ¼ ¼ ¼ ¼ ¼ j=1 i=j+1 2 n-1 1 an an ¼ an ä ä I M This determinant is called the Vandermonde determinant. http://functions.wolfram.com 18 Complex characteristics Real part f x + ä y + f x - ä y Re f x + ä y 2 H L H L H H LL 1 y2 y2 Re f x + ä y f x - x - + f x + x - 2 x2 x2 H H LL Imaginary part f x + ä y - f x - ä y Im f x + ä y 2 ä H L H L H H LL x y2 y2 y2 Im f x + ä y - f x - x - - f x + x - 2 y x2 x2 x2 H H LL Absolute value f x + ä y f x + ä y f x - ä y H L¤ H L H y2 L y2 f x + ä y f x - x - f x + x - x2 x2 H L¤ Argument f x + ä y + f x - ä y f x + ä y - f x - ä y arg f x + ä y tan-1 , 2 2 ä H L H L H L H L H H LL 1 y2 y2 x y2 y2 y2 arg f x + ä y tan-1 f x - x - + f x + x - , - f x - x - - f x + x - 2 x2 x2 2 y x2 x2 x2 H H LL Conjugate value f x + ä y f x - ä y f z f z H L H L 1 y2 y2 ä x y2 y2 y2 f HxL+ ä yH L f - x + x + f x - x - - - f x - x - - f x + x - 2 x2 x2 2 y x2 x2 x2 H L http://functions.wolfram.com 19 Differentiation Low-order differentiation Derivatives of the first order f z + Ε - f z f ¢ z lim Ε®0 Ε This limit definesH L the HderivativeL of a function f at the point z, if it exists. H L ¶ c f z ¶ f z c ¶z ¶z ThisH formulaH LL reflectsH L the property that a constant factor can be pulled out of the differentiation. ¶ f z ± g z ¶ f z ¶g z ± ¶z ¶z ¶z ThisH H formulaL H LL reflectsH L the propertyH L that the derivative of a sum (and difference) is equal to the sum (and difference) of the derivatives. ¶ f z g z ¶ f z ¶g z g z + f z ¶z ¶z ¶z TheH H productL H LL ruleH forL differentiationH L shows that the derivative of a product is equal to the derivative of the first H L H L function multiplied by the second function plus the derivative of the second function multiplied by the first function. ¶ f z g z h z ¶h z ¶ f z ¶g z f z g z + h z g z + f z h z ¶z ¶z ¶z ¶z TheH H productL H L H ruleLL for HdifferentiationL showsH L how to evaluateH L the derivative of the product of three functions. H L H L H L H L H L H L n n n ¶ k=1 fk z 1 ¶ fj z fk z ¶z j=1 fj z ¶z k=1 IÛ H LM H L The product ruleâ for differentiationä H showsL how to evaluate the derivative of the product of n functions. H L ¶ f z 1 ¶ f z ¶g z g z - f z ¶z g z g z 2 ¶z ¶z H L H L H L The quotient rule HforL differentiationH L shows that the derivative of the ratio is equal to the derivative of the numerator multipliedH L byH Lthe denominator minus the derivative of the denominator multiplied by the numerator, divided by the square of the denominator. f z h z ¶ ¢ ¢ ¢ g z g z h z f z - f z h z g z + f z g z h z ¶H Lz H L g z 2 H L H L H L H L H L H L H L H L H L H L H L http://functions.wolfram.com 20 The quotient rule for differentiation has been generalized to the case when the numerator is the product of two functions. f z ¶ ¢ ¢ ¢ g z h z g z h z f z - f z h z g z - f z g z h z ¶zH L g z 2 h z 2 H L H L H L H L H L H L H L H L H L H L H L The quotient rule for differentiation is generalized to the case when the denominator is the product of two functions. H L H L ¥ ¥ ¶ ¶ak z ak z ¶z k=0 k=0 ¶z H L Thisâ formulaH L âshows that the derivative of the sum is equal to the sum of the derivatives. For an infinite sum it is true under some restrictions on ak z , which ensure the convergence of the series. ¶ f z ¥ ¥ a k zk-1 ; z < r f z a zk ; z < r k H L k ¶z k=1 k=0 H L This formulaâ shows that ¤ theí derivativeH L â of a power ¤ series is equal to the corresponding sum of the derivatives. It is true inside the corresponding circle of convergence with radius r. ¶ f g z f ¢ g z g¢ z ¶z ThiHs HchainLL rule for differentiation shows that the derivative of composition f g z is equal to the derivative of the H H LL H L outer function f in the point g z , multiplied by the derivative of the inner function g. ¶ f g z , h z H H LL g¢ z f 1,0 g z , h z + h¢ z f 0,1 g z , h z ¶z H L H L H L ThisH HchainL H LruleL for partial differentiation generalizes the previous chain rule for differentiation in the case of a H L H H L H LL H L H H L H LL function with two variables f u, v ; u g z v h z . n ¶ f g1 z , g2 z , ¼, gn z g ¢ z f 0,¼,1,¼,0 g z , ¼, g z , ¼, g z H k L H L ì1 H kL n ¶z k=1 H L H H L H L H LL This chain rule for partial âdifferentiationH L generalizesH H L theH chainL ruleH L Lfor differentiation in the case of a function with several variables f u1, u2, ¼, un ; uk gk z 1 £ k £ n. ¶ f -1 z 1 ¶z f ¢ f -1 Hz L H L ì H L H L This formula showsH L that the derivative of the inverse function f -1 z is equal to the reciprocal of the derivative of the direct functionI HfL inM the point f -1 z . H L ¶ f z H L H L ¶ 2 ¶z ¶ f z H L ¶zH L ¶z2 H L http://functions.wolfram.com 21 This formula shows that the composition of the first derivatives is equal to the derivative of the second order. ¶ f z â z f z ¶z This formula shows that the derivative of an indefinite integral produces the original function (the derivative is the à H L H L inverse operation to the indefinite integration). ¶ z f t ât f z ¶z a This formula shows that the derivative of a definite integral with respect to the upper limit produces the original à H L H L function. ¶ b f t ât - f z ¶z z This formula shows that the derivative of a definite integral with respect to the low limit gives the original function à H L H L with a negative sign. ¶ b b f t, z ât f 0,1 t, z ât ¶z a a H L This formula shows that the order of differentiation and definite integration can be changed if the limits of the à H L à H L integral do not depend on the variable of differentiation. ¶ h z h z f t, z ât f 0,1 t, z ât - f g z , z g¢ z + f h z , z h¢ z ¶z g z g z H L H L H L This formula reflects the general rule of differentiating an integral when its limits and its integrand depend on the à H L H L à H L H L H H L L H L H H L L H L variable of differentiation. Derivatives of the second order ¶2 f z f z - 2 f z + Ε + f z + 2 Ε lim ¶z2 Ε®0 Ε2 ThisH limitL definesH L the secondH L derivativeH L of a function f at the point z, if it exists. ¶2 f g z f ¢¢ g z g¢ z 2 + f ¢ g z g¢¢ z ¶z2 H H LL This formula showsH H LL howH L to evaluateH H LL HtheL second derivative of a general composition f g z . Symbolic differentiation H H LL Definition 1 n n f n z lim -1 n-k f z + k Ε Ε®0 n k Ε k=0 H L H L âH L H L http://functions.wolfram.com 22 This limit defines the nth-order derivative of a function f at the point z, if it exists. Converting to finite differences and back ¶n f z ¥ h-n = S n Dn+k f z ; Dk f z Dk-1 D f z D f z f z + h - f z n k+n z,h z,h z,h z,h z,h ¶z k=0 n + 1 k H L H L ¥ hj+n ¶k+n f z n â n H L k H L k-H1 H LL í H L H L H L Dz,h f z = H L Sk+n ; Dz,h f z Dz,h Dz,h f z Dz,h f z f z + h - f z k+n k=0 n + 1 k ¶z H L H L ProdHucL ts â H L H H LL í H L H L H L H L n n ¶k f z ¶n-k g z ¶ z,n f z g z k k n-k k=0 ¶z ¶z 8 < H L H L th ThisH ruleH L isH LcalledL â the binomial differentiation rule for the n -order derivative. 3 n n n n 3 nk ¶ k=1 fk z ¶ fk z ∆n,n1+n2+n3 n1 + n2 + n3; n1, n2, n3 ¶zn ¶znk n1=0 n2=0 n3=0 k=1 IÛ H LM H L This rule is calledâ theâ multinomialâ differentiationH rule for theLä nth-order derivative. n m n n n m nk ¶ ¶ fk z fk z ¼ ∆n, m n n1 + n2 + ¼ + nm; n1, n2, ¼, nm ¶zn k=1 k ¶znk k=1 n1=0 n2=0 nm=0 k=1 Ú H L Thisä rule HisL calledâ âthe multinomialâ differentiationH rule for the nth-orderLä derivative. Ratios n f z ¶ n n-k k j -j-1 k j g z ¶ f z -1 k + 1 g z ¶ g z n! ; n Î N nH L n-k k ¶z k=0 ¶z j=0 j + 1 ! n - k ! k - j ! ¶z H L H L H L H L HthL H L This formulaâ shows howâ to evaluate an n -order derivative for the quotient of two functions. H L H L H L n n f z ¶ f z ¶ n n-k k j k j g z ¶zn ¶ f z -1 k + 1 ¶ g z + n! g z -j-1 ; n Î N nH L H L n-k k ¶z g z k=1 ¶z j=1 j + 1 ! n - k ! k - j ! ¶z H L H L H thL H L H L This formula showsâ how to evaluateâ an n -order derivativeH Lfor the quotient of two functions. H L H L H L H L n n f z ¶ f z ¶ n n-m m q g z ¶zn ¶ f z -1 m + 1 + n! g z -q-1 nH L H L n-m ¶z g z m=1 ¶z q=1 q + 1 ! n - m ! m - q ! H L H qL-2 H L H L q-1 m- k j q-1 q-1 m- k j m mâ-k 1 j=1â m - i-1 k j ¶Hk iL g z ¶ j=1 g z H L ¼ H L H j=1 L H L ; n Î N k p ¶zk i m- q-1k j k 1 =0 k 2 =H 0L k qÚ-1 =H0 L i=1 i=1 H L ¶zÚ jH=1L H L H L th Ú H L H L This formula showsâH L âH howL to HevaluateâL ä an n -order derivativeä for the quotientÚ ofH Ltwo functions. H L http://functions.wolfram.com 23 n n f z ¶ f z ¶ n n-k g z ¶zn ¶ f z + n! nH L H L n-k ¶z g z k=1 ¶z H L k -1 m k +H 1L k k k m ¶nk g z â -m-1 g z ¼ ∆ m n1 + n2 + ¼ + nm; n1, n2, ¼, nm ; n Î N H L k, nk n m + 1 ! n - k ! k - m ! k=1 ¶z k m=1 n1=0 n2=0 nm=0 k=1 H L H L Ú H L This formulaâ shows how to evaluate anH nL th-orderâ âderivativeâ for the Hquotient of two functions. Lä H L H L H L Power a k ¶n f z n - a n -1 k n ¶n f z a f z a-k ; n Î N n n k n ¶z k=0 a - k ¶z H L H L th H L a This formula showsâ how to evaluateH Lan n -order derivative of the power f z . n a n m n m ¶ f z a n - a -1 n a-m ¶ f z ∆n f z + a f z ; n Î N n n m n H L ¶z m=1 a - m ¶z H L H L th H L a This formula showsH L how to evaluateâ an n -orderH L derivative of the power f z . n a ¶ f z a f z ∆n + ¶zn H L m-2 m-1 H L n m n n-k 1 n- j=1 k j m-1 p-1 m-1 k i n- k j n - a -1 n n - k j ¶ f z ¶ j=1 f z a H L f z a-m ¼ j=1 ; n Î N n a - m m k p ¶zk i n- m-1k j m=1 k 1 =0 k 2 =H 0L k Úm-1 =H0L p=1 i=1 H L ¶zÚ j=H1L H L H L H L th Ú H L a H L This formula showsâ how to evaluateH L anâH L n âH -orderL derivativeH âL ä of the power f z ä. Ú H L H L n a n m n n n m nk ¶ f z a n - a -1 n a-m ¶ f z ∆n f z + a f z ¼ ∆n, m n n1 + n2 + ¼ + nm; n1, n2, ¼, nm ; ¶zn n a - m m k=1 k H L ¶znk m=1 n1=0 n2=0 nm=0 k=1 n ÎH NL H L Ú H L H L â H L â â â H Lä This formula shows how to evaluate an nth-order derivative of the power f z a. Positive integer powers H L 2 ¶n f z n n f n-k z f k z ; n Î N n k ¶z k=0 H L H L H L 2 This formulaâ shows howH L to evaluateH L an nth-order derivative of the square f z . 3 ¶n f z n n-k1 n n - k1 f k1 z f n-k1-k2 z f k2 z ; n Î N n H L ¶z k1 k2 k1=0 k2=0 H L H L H L H L This formulaâ showsâ how to evaluateH Lan nth-orderH L derivativeH L of the square f z 3. H L http://functions.wolfram.com 24 3 ¶n f z n n n 3 ¶ki f z ∆ k + k + k ; k , k , k ; n Î N n n,k1+k2+k3 1 2 3 1 2 3 ¶z ki k1=0 k2=0 k3=0 i=1 ¶z H L H L This formulaâ showsâ â how to evaluateH an nth-order derivativeLä of the cube f z 3. 4 ¶n f z n n-k1 n-k1-k2 n n - k1 n - k1 - k2 f k1 z f k2 z f n-k1-k2-k3 z f k3 z ; n Î N n H L ¶z k1 k2 k3 k1=0 k2=0 k3=0 H L H L I M I M H L This formulaâ showsâ â how to evaluate an nth-order derivativeH L ofH Lthe fourth powerH L fH Lz4. 4 ¶n f z n n n n 4 ¶ki f z ∆ k + k + k + k ; k , k , k , k ; n Î N n n,k1+k2+k3+k4 1 2 3 4 1 2 3 4 H L ¶z ki k1=0 k2=0 k3=0 k4=0 i=1 ¶z H L H L This formulaâ showsâ â howâ to evaluate Han nth-order derivative of theLä fourth power f z 4. m-2 m-1 n m n n-k 1 n- j=1 k j m-1 p-1 m-1 k i n- k j ¶ f z n - k j ¶ f z ¶ j=1 f z ¼ j=1 ; mHÎLN+ n Î N n m-1 ¶z k p ¶zk i n- k j k 1 =0 k 2 =H 0L k Úm-1 =H0L p=1 i=1 H L ¶zÚ j=H1L H L H L H L Úth H L H L m + This formulaâH L showsâH L howH âto Levaluateä an n -order derivativeä of the generalÚ HintegerL powerì f z ; m Î N . H L m ¶n f z n n n m ¶nk f z + ¼ ∆n, m n n1 + n2 + ¼ + nm; n1, n2, ¼, nm ; m Î N n Î N ¶zn k=1 k ¶znk H L n1=0 n2=0 nm=0 k=1 H L Ú H L m This formulaâ showsâ howâ to evaluateH an nth-order derivative of Ltheä general integer powerì f z ; m Î N+. Negative integer powers H L n 1 ¶ n k n k f z -1 n ¶ f z n + 1 f z -k-1 ; n Î N+ n k n ¶z k=0 k + 1 ¶z H L H L HthL This formulaH showsL â how to evaluateH L an n -order derivative of the reciprocal 1 f z . n 1 ¶ n k n k f z ∆n -1 n ¶ f z + n + 1 f z -k-1 ; n Î N H L n k n ¶z f z k=1 k + 1 ¶z H L H L th H L This formula showsH howL â to evaluate HanL n -order derivative of the reciprocal 1 f z . H L n 1 m-2 m-1 ¶ n m n n-k 1 n- j=1 k j m-1 p-1 m-1 k i n- k j f z ∆n -1 n n - k j ¶ f z ¶ j=1 f z + n + 1 f z -m-1 ¼ j=1 H L ; n m-1 ¶z f z m + 1 m k p ¶zk i n- k j m=1 k 1 =0 k 2 =H 0L k Úm-1 =H0L p=1 i=1 H L ¶zÚ j=H1L H L H L H L H L n Î N Ú H L H L H L â H L âH L âH L H âL ä ä Ú H L H L H L This formula shows how to evaluate an nth-order derivative of the reciprocal 1 f z . H L http://functions.wolfram.com 25 n 1 ¶ n m n n n m n f z k ∆n -1 n -m-1 ¶ f z + n + 1 f z ¼ ∆n, m n n1 + n2 + ¼ + nm; n1, n2, ¼, nm ; n Î N ¶zn f z m + 1 m k=1 k ¶znk m=1 n1=0 n2=0 nm=0 k=1 H L H L Ú H L This formula showsH howL â to evaluate anH Lnth-orderâ â derivativeâ of the Hreciprocal 1 f z . Lä H L Log from function H L k ¶n log f z n -1 k-1 n ¶n f z ∆n log f z + ; n Î N n k k n ¶z k=1 k f z ¶z H H LL H L th H L This formula showsH howH LL toâ evaluate an n -order derivative of the composition with logarithm log f z . H L ¶n log f z ¶zn H H LL m-2 m-1 H H LL n m-1 n n-k 1 n- j=1 k j m-1 m-1 k i n- k j -1 n p-1 ¶ f z ¶ j=1 f z -m n - j=1 k j ∆n log f z + f z ¼ ; n Î N m m k p ¶zk i n- m-1k j m=1 k 1 =0 k 2 =H 0L k Úm-1 =H0L p=1 i=1 H L ¶zÚ j=H1L H L H L H L th Ú H L H L This formulaH H L Lshowsâ how to evaluateH L an nâH L -orderâH L derivativeH âL ofä the composition withä logarithm log Úf z H .L H L H H LL This formula shows how to evaluate an nth-order derivative of the composition with logarithm log f z . Exp from function H H LL ¶n ãg z n 1 m m ¶n g z m-j ãg z -1 j g z j ; n Î N ¶zn m! j ¶zn H L m=0 j=0 H L thH L This formula showsâ âhowH toL evaluateH Lan n -order derivative of the composition with exponential function exp f z . ¶n ãg z n 1 m ¶n g z m-j g z j m j ã ∆n + -1 g z ; n Î N ¶zn m! j ¶zn H H LL H L m=1 j=0 H L th H L This formula showsâ how toâ evaluateH L an nH L-order derivative of the composition with exponential function exp f z . ¶n ãg z n n! n g z j k j + ã ∆ n g z ; n Î N n m, kr ¶z r=1 n k j H H LL H L m=1 k1+ 2 k2+¼+ n kn=n j=1 j! kj ! j=1 H L H L Ú This formula showsâ howâ to evaluate an nth-order derivativeä H L of the composition with exponential function exp f z . Û ¶n ãg z n 1 m m ãg z -1 q g z q ¶zn m! q H H LL H L m=0 q=0 H L m-q-2 n- k j n- m-q-1k j â âH L HnL n-k 1 j=1 m-q-1 p-1 m-q-1 ¶k i g z ¶ j=1 g z n n - j=1 k j ∆n ∆m-q - g z ∆m-q + ¼ ; n Î N k p ¶zk i n- m-q-1k j k 1 =0 k 2 =H 0L k mÚ-q-1 =H0L p=1 i=1 H L ¶zÚ j=1 H L H L H L H L Ú H L H L H L âH L âH L H â L ä ä Ú H L H L http://functions.wolfram.com 26 This formula shows how to evaluate an nth-order derivative of the composition with exponential function exp f z . ¶n ãg z n 1 m n n n m-j ¶nk f z g z j m j ã ∆ + -1 g z ¼ ∆ m-j n + n + ¼ + n ; n , n , ¼, n ; n n, n 1 2 m-j 1 2 m-j H H LL ¶zn m! j k=1 k ¶znk H L m=1 j=0 n1=0 n2=0 nm-j=0 k=1 H L n Î N Ú H L â âH L H L â â â I M ä This formula shows how to evaluate an nth-order derivative of the composition with exponential function exp f z . Function from power H H LL n a n k j k a ¶ f z -1 a k - a j - n + 1 n f z ; n Î N ¶zn n-a k k=0 j=0 j! k - j ! z H L H L H L H L H L th a This formulaâ âshows how to evaluate an n -order derivative of the composition with power function f z . H L n 2 n ¶ f z 2 k - n + 1 2 n-k f k z2 ; n Î N n n-2 k H L ¶z k=0 n - k ! 2 z H L H L I M H L th 2 This formulaâ shows how to evaluateI M an n -order derivative of the composition f z . H L H L n 1 ¶ f 1 n n - 1 ! 1 z n n k f ∆n + -1 f ; n Î N I M n k+n k ¶z z k=1 k - 1 ! z z J N H L H L th This formula showsH howL âto evaluate the derivative of the n -order of the composition f 1 z . H L ¶n f z n n-k -1 k 2 n-k f k z ; n Î N H L ¶zn 2 n-k k=0 n - k ! 2 z H L I N H L H L H L â I N This formula shows how to evaluate an nth-order derivative of the composition f z . H L I N Function from exponent I M ¶n f ãz n ãk z S k f k ãz ; n Î N+ n n ¶z k=0 H L H L H L th z This formulaâ shows howH toL evaluate an n -order derivative of the composition f ã . ¶n f az n logn a ak z S k f k az ; n Î N n n H L ¶z k=0 H L H L H L th z This formula showsH L â how to evaluateH L an n -order derivative of the composition f a . n n - 1 ! ¶n f ã1 z n k k 1 z n m z m m 1 z H L f ã ∆n + -1 ã Sk f ã ; n Î N ¶zn k+n k=1 k - 1 ! z m=0 H L H L H L I M I M H L â â I M H L http://functions.wolfram.com 27 This formula shows how to evaluate an nth-order derivative of the composition f ã1 z . n k n b z n - 1 ! b log c ¶ f c n k k m b I M b z n z m m b z f c ∆n + -1 c Sk f c ; n Î N ¶zn k+n k=1 k - 1 ! z m=0 H L H H LL H L H L I M th b z This formulaI showsM howH Ltoâ evaluate an n -order derivativeâ of the compositionI M f c . H L ¶n f ã z n n-k k -1 k 2 n-k m z m m z ã Sk f ã ; n Î N I M ¶zn 2 n-k k=0 n - k ! 2 z m=0 H L H L J N H L H L H L â â J N This formula shows how to evaluate an nth-order derivative of the composition f ã z . H L I N n z2 n k ¶ f ã 2 k - n + 1 2 n-k m z2 m m z2 ã Sk f ã ; n Î N J N n n-2 k ¶z k=0 n - k ! 2 z m=0 H L H L H L I M H L 2 This formulaâ shows how to evaluateâ an nth-orderJ derivativeN of the composition f ãz . H L H L ¶n f ãza n k -1 j a k - a j - n + 1 k n m za m m za ã Sk f ã ; n Î N n n-a k I M ¶z k=0 j=0 j! k - j ! z m=0 H L H L H L H L I M a This formulaâ showsâ how to evaluate an nâth-order derivativeI M of the composition f ãz . H L ¶n f cb z n n-k k k -1 k 2 n-k b log c m b z m m b z c Sk f c ; n Î N I M ¶zn 2 n-k k=0 n - k ! 2 z m=0 H L H L J N H L H L H H LL H L â â J N This formula shows how to evaluate an nth-order derivative of the composition f cb z . H L I N n b z2 n k k ¶ f c 2 k - n + 1 2 n-k b log c m b z2 m m b z2 c Sk f c ; n Î N J N n n-2 k ¶z k=0 n - k ! 2 z m=0 H L H L H L I M H L H H LL 2 This formulaâ shows how to evaluate an nthâ-order derivativeJ of Nthe composition f cb z . H L H L ¶n f cb za n k -1 j -a j + a k - n + 1 b log c k k n m b za m m b za c Sk f c ; n Î N n n-a k I M ¶z k=0 j=0 j! k - j ! z m=0 H L H L H L H H LL H L I M a This formulaâ showsâ how to evaluate an nth-order derivativeâ of the compositionI M f cb z . H L Function from trigonometric functions I M http://functions.wolfram.com 28 ¶n f sin z f sin z ∆n + ¶zn m-j H nH LL1 m-1 m ä m - j H H LL -1 j 2j-m sinj z j + 2 l - m n exp - n Π - j + 2 l - m Π - 2 z f m sin z ; n Î N j l m=1 m! j=0 l=0 2 H L th This formulaâ âshows howâH to Levaluate anH L nH -order derivativeL ofH the compositionH L H f sinLLz . H H LL ¶n f sin z n f m sin z ¶n sin y - sin z m . y ® z ; n Î N+ ¶zn m! ¶yn H H LL m=1 H L H H LL H H LL H H L thH LL This formula âshows how to evaluate an n -order 8derivative< of the composition f sin z . ¶n f cos z n 1 m-1 m-j n j m j-m j n j+2 l-m ä z m - j m f cos z ∆n + ä -1 2 cos z j + 2 l - m ã f cos z ; n Î N n j H H LL l ¶z m=1 m! j=0 l=0 H L H L H H LL th This formula showsH H LhowL to evaluateâ âan Hn -orderL derivativeâ of HtheL H compositionL f cos z . H H LL ¶n f cos z n f m cos z ¶n cos y - cos z m . y ® z ; n Î N+ ¶zn m! ¶yn H H LL m=1 H L H H LL H H LL H H L th H LL This formula showsâ how to evaluate an n -order derivative8 < of the composition f cos z . Function from hyperbolic functions H H LL ¶n f sinh z ¶zn m-j H H LL n 1 m-1 m m - j f sinh z ∆ + -1 n -1 j-l 2j-m sinhj z j + 2 l - m n ã- j+2 l-m z f m sinh z ; n Î N+ n j l m=1 m! j=0 l=0 H L H L th This HformulaH LL showsH Lhowâ to evaluateâ anâ nH -orderL derivativeH L H of the compositionL f sinh z . H H LL ¶n f sinh z n ¶n sinh y - sinh z m f m sinh z . y ® z ; n Î N+ ¶zn ¶yn m! H H LL m=1 H L H H LL H H L H LL th H H LL This formula showsâ how to evaluate an n 8 -order< derivative of the composition f sinh z . ¶n f cosh z n 1 m-1 m-j j m j-m j n 2 i+j-m z m - j m f cosh z ∆n + -1 2 cosh z 2 i + j - m ã f cosh z ; n Î N n j H H LL i ¶z m=1 m! j=0 i=0 H L H L H H LL th This formula showsH howH LL to evaluateâ âan nH -orderL derivativeâ of HtheL H compositionL f cosh z . H H LL ¶n f cosh z n f m cosh z ¶n cosh y - cosh z m . y ® z ; n Î N+ ¶zn m! ¶yn H H LL m=1 H L H H LL H H LL H H thL H LL This formula showsâ how to evaluate an n -order derivative 8 of< the composition f cosh z . General compositions H H LL http://functions.wolfram.com 29 ¶n f g z n 1 m m ¶n g z m-j -1 j g z j f m g z ; n Î N n j n ¶z m=0 m! j=0 ¶z H L H H LL thH L This formulaâ showsâ howH toL evaluateH L an n -order derivativeH H LL of the general composition f g z . ¶n f g z n 1 m ¶n g z m-j j m j m f g z ∆n + -1 g z f g z ; n Î N n j n H H LL ¶z m=1 m! j=0 ¶z H L H H LL th H L This formula HshowsH LL howâ to evaluateâH anL n -orderH L derivative of theH H LgeneralL composition f g z . ¶n f g z n f m g z n! n j k j + ∆ n g z ; n Î N n m, kr ¶z r=1 n k j H H LL m=1 k1+ 2 k2+¼+ n kn=n H j=L1 j! kj ! j=1 H L H H LL Ú H H LL This formulaâ is calledâ Faá di Bruno's formula. ä H L Û ¶n f g z n 1 m q m q n -1 g z ∆n ∆m-q - g z ∆m-q + n q ¶z m=0 m! q=0 H L H H LL m-q-2 m-q-1 â n nâ-k 1H Ln- j=1 kHj L m-q-1 p-1H L m-q-1 k i n- k j n - k j ¶ g z ¶ j=1 g z ¼ j=1 f m g z ; n Î N k p ¶zk i n- m-q-1k j k 1 =0 k 2 =H 0L k mÚ-q-1 =H0L p=1 i=1 H L ¶zÚ j=1 H L H L H L H L Ú H L HthL This formula, FaáâH L diâH L Bruno'sH â relationL ä, shows how to evaluateä an n -order derivativeÚ H L of theH HgeneralLL composition H L f g z . ¶n f g z f g z ∆n + H ¶HznLL H HnLL 1 m n n n m-j ¶nk f z m j m j f g z -1 g z ¼ ∆ m-j n + n + ¼ + n ; n , n , ¼, n ; n Î N H H LL n, n 1 2 m-j 1 2 m-j m! j k=1 k ¶znk m=1 j=0 n1=0 n2=0 nm-j=0 k=1 H L Ú H L This formulaâ showsH H L LhowâH to LevaluateH anL â nth-orderâ derivativeâ ofI the general composition f g z . M ä General power exponential compositions H H LL ¶n f z g z ¶zn H L H L n 1 m-1 n n n g z j m j f z ∆n + -1 g z log f z ¼ ∆ m-j k 1 + k 2 + ¼ + k m - j ; k 1 , k j n, v=1 k v m=1 m! j=0 k 1 =1 k 2 =0 k m-j =0 H L m-j k i s Ú -1H Lh-1 s s s H L â âH L H H L H H LLL kâH Li âH L H âL H H Ls H L H L H L H 2 , ¼, k m - j ∆ log f z + ¼ ∆ h s s, q u s h h u=1 i=1 s=H 0L h=1 h f z q 1 =0 q 2 =0 q h =0 H L H L h q i k i -sÚ H L L H LL H H LL ¶ f z ¶ g z äâq 1 + q 2 + ¼ + q h ; qâ1 , q 2 , ¼, q h âH L âH L âH L ; n Î N H L ¶zq i ¶zk i -s i=1 H L H L H L H L g z This formula shows how to evaluate an nth-orderH H LderivativeH L of theH LgeneralH L HpowerL exponentialH LLä compositionH L H L f z . H L H L http://functions.wolfram.com 30 ¶n zz ¶zn n 1 m-1 n n n z j m j z ∆n + -1 z log z ¼ ∆ m-j k 1 + k 2 + ¼ + k m - j ; k 1 , k 2 , ¼, k m - j j n, v=1 k v m=1 m! j=0 k 1 =0 k 2 =0 k m-j =0 m-j Ú H L â âH L H 1 H LL âH 1L 1âH-Lk i H âL 0 H H L 0 H L H L H L H L H LL k i Sk i -1 + Sk i z + log z k i Sk i -1 + z Sk i ; n Î N i=1 H L H L H L H L H L H L thH L H L H L z This formula shows howä to IevaluateI H L an n -orderM derivativeH L I H ofL the general MpowerM exponential composition z . Inverse function j n n n n i -1 i n -n n 1 f f z -1 -1 ¢ -1 i=2 ji f z ∆n f z + f f z ¼ ∆ n i-1 j ,n-1 -1 n + ji - 1 ! ; n Î N i=2 i ¢ -1 ji ! i! f f z j2=0 jn=0 i=2 i=2 H L H L H LH L H L H L Ú Ú H L I H LM This formulaH L showsH L howI Ito evaluateH LMM âan nthâ-order derivativeH ofL the inverseâ functionä f -1 z . H L I H LM Repeated derivatives H L H L ¶ ¶ n ¶k f z k k z ¼ z f z Sn z ; n Î N k ¶z ¶z k=0 ¶z n times H L H L H H LL â This formula shows how to apply n times the operation z ¶ to the function f z . ¶z ¶ ¶ n ¶k f z k k z - a ¼ z - a f z Sn z - a ; n Î N k H L ¶z ¶z k=0 ¶z n times H L H L H L H L H H LL â H L This formula shows how to apply n times the operation z - a ¶ to the function f z . ¶z Fractional integro-differentiation H L H L f t z-t n+Α-1 ¶n z ât ¶Α f z 0 G n+Α n n Re -Α + 1 Re -Α £ 0 H L H¶z L Α ¶z z f t z-Ht -ΑL-1 Ù ât True 0 G -Α H L d H Lt ß H L th H L H L The Α fractionalH integroL -derivative of the function f z with respect to z is defined by the preceding formula, Ù where the integration in Mathematica should be performed with the option GenerateConditions->False: f t z t Α+n-1 Integrate - , t, 0, z , GenerateConditions ® False. This definition supports the Gamma Α+n H L Riemann-Liouville@ D-HadamardH L fractional left-sided integro-differentiation at the point 0. @ D ¶Α f z z fBt z - t -Α-1 8 < ât ; Re -Α > 0 ¶zΑ 0 G -Α H L H L H L à H L H L http://functions.wolfram.com 31 This formula for the Αth fractional integro-derivative represents the fractional integral of the function f z with respect to z. This integral is called the Abel integral. z f t z-t n+Α-1 H L ¶n ât ¶Α f z 0 G n+Α ; n Re -Α + 1 Re -Α £ 0 ¶zΑ H L¶H znL H L KÙ O ThisH formulaL for the Αth fractional integro-derivative actually represents the fractional derivative of the function d H Lt ß H L f z with respect to z. This derivative includes the composition of the corresponding usual nth derivative of order n Re -Α + 1 and an Abel integral. Α ¥ k ¥ k-Α ¶H L k=0 ck z k! ck z d ΑH Lt ¶z k=0 G k - Α + 1 IÚ M th This formula showsâ how to evaluate the Α fractional integro-derivative of the analytical function near the point z 0. H L Α n ¥ a+k ¥ ¶ log z k=-¥ ck z Α k+a-Α ck FC z, a + k, n z ; n Î N Α log ¶z k=-¥ H L I H L Ú M th This formula shows how âto evaluateH the Α fractionalL integro-derivative of a function having Laurent series expansion, multiplied on logn z za near the point z 0. j + k k! a -z j zk-Α ¶Α f z ¥ ¥ j+k k 0 ¥ H L k ; f z ak z - z0 Α ¶z k=0 j=0 G k - Α + 1 k=0 H L H L th This formulaââ shows that for the evaluation HofL theâ Α HfractionalL integro-derivative of the analytical function f z H L near the point z z0, you need to re-expand this function in a series near the point z 0 and then evaluate the corresponding integro-derivative. H L Integration Indefinite integration For the direct function f ¢ z â z f z Integration of the derivative gives the original function. à H L H L f z ± g z â z f z â z ± g z â z The integral of the sum gives the sum of the integrals. à H H L H LL à H L à H L c f z â z c f z â z à H L à H L http://functions.wolfram.com 32 The constant factor can be placed outside of the integral. f g z g¢ z â z f w âw ; w g z This formula reflects the changing variables rule in the integral. à H H LL H L à H L H L f z g¢ z â z f z g z - g z f ¢ z â z This formula reflects the integration by parts rule. à H L H L H L H L à H L H L ¶n g z ¶n f z n-1 ¶k f z ¶n-k-1 g z f z â z -1 n g z â z + -1 k n n k n-k-1 ¶z ¶z k=0 ¶z ¶z H L H L H L H L àThisH formulaL reflectsH theL à generalizedH L integrationâH L by parts rule. ¥ ¥ f z â z uk z â z ; f z uk z k=0 k=0 àTheH integralL â ofà the HsumL is equalH L toâ theH sumL of the integrals of the summands (under some restrictions for conver- gence of the occurring infinite series). ¥ a zk+1 ¥ k k f z â z ; z < r f z ak z ; z < r k=0 k + 1 k=0 àTheH integralL â from the power ¤ seriesí H isL equalâ to the sum¤ of the integrals from each term of the series (inside some circle of convergence). Repeated indefinite integration z z - t n-1 â z ¼ â z f z f t ât 0 n - 1 ! n times H L à Kà H H LL à H L This formula reflects repeatedH indefiniteL integration, where the integration in Mathematica should be performed z t n-1 with the option GenerateConditions -> False: Integrate - f t , t, 0, z , n-1 ! GenerateConditions False ® . H L H L n-1 @ @ D 8 < 1 1 z log z - log t â z ¼ â z f z f t ât z z 0 t Dn - 1 ! n times H H L H LL à à H H LL à H L This formula reflects repeated indefiniteH integration,L where the integration in Mathematica should be performed Log z -Log t n-1 with the option GenerateConditions -> False: Integrate f t , t, 0, z , t n-1 ! GenerateConditions ® False . H @ D @ DL H L @ @ D 8 < Definite integration D http://functions.wolfram.com 33 For the direct function b ¥ b - a 2 k+1 b - a f t ât f 2 k k a j=0 4 2 k + 1 ! 2 H L H L b à ¢H L â f t ât f b -H f a L a b b b à fHtL ± g t âH tL H Lf t ât ± g t ât a a a b b à cH fH Lt âtHLLc fàt âtH L à H L a a ¢ ¢ à f z HgL z â z àf z HgL z - g z f z â z b g b ¢ à fH Lg t H gL t ât H L H L f àΤ âHΤL ; ΤHL g t a g a H L a b à f Ht H âLLt H-L f àt âH tL H L H L b a b c b à f HtL ât à f t H âL t + f t ât a a c b c c à f HtL ât + à f Ht L ât à f HtL ât a b a ¥ k+1 k+1 ¥ b ak b - a H L H L H L k à f t ât à à ; f z ak z a k=0 k + 1 k=0 I M a H L b H L à f t g t âât f Τ g t ât ; Τ Î a, bâ g t ³ 0 ; t Î a, b b a a b à f HtL gHtL ât ΜH Làg t HâLt ; min f Ht £LΜì£ max@ D f t H L b a This formula reflects the first mean value theorem. à H L H L à H L H H LL H H LL a Α f t g t ât f a g t ât ; Α Î a, b f ¢ t < 0 f t ³ 0 ; t Î a, b b a This formula reflects the second mean value theorem. à H L H L H Là H L H L ì H L ì H L H L a b f t g t ât f b g t ât ; Β Î a, b f ¢ t > 0 f t ³ 0 ; t Î a, b b Β a Α b à f HtL gHtL ât f HaLà gHtL ât + f b H gL ìt âtH ;LΑ Î ìa, bH L f ¢ t > 0H ; t ÎL a, b b a Α à H L H L H Là H L H Là H L H L ì H L H L http://functions.wolfram.com 34 Β b b Β f t, z ât â z f t, z â z ât Α a a Α b b f t, Τ b b f t, Τ à à H L âàt âΤà H L âΤ ât - Π2 f x, x ; x Î a, b a a t - x Τ - t a a t - x Τ - t This formulaH isL called the Poincaré-BertrandH L formula. à à à à H L H L H L H L H L H L Β z Β Β f t, z ât â z f t, z â z ât 0 0 0 t b ¥ ¥ b H L H L à à ak z â z à àak z â z a k=0 k=0 a àOrthâogonH aLlity âà H L See Generalized Fourier series in the section Series representations. Cauchy integrals 1 j t F z ât 2 Π ä L t - z This formula is HcalledL the Cauchy-type integral along the piecewise smooth contour L (which can be closed or H L à opened). The density function j Τ must be continuous along L, but it must also meet a more stringent test known as the Hölder condition. The function j t satisfies the Hölder condition if, for two arbitrary points t1, t2 on the Λ curve, j t2 - j t1 < A t2 - t1 for some positive constants A and 0 < Λ £ 1. The Cauchy-type integral is analytic H L everywhere on the complex plane except on the contour L itself, which is a singular line for this integral. Since the H L integral contains a factor (called the kernel) in the form of 1 Τ - z , it diverges at Τ z for any z lying on L. H L H L¤ ¤ 1 j t j z z Î D+ ât - 2 Π ä L t - z 0 z Î D H L This formulaH L is calledH L the Cauchy integral formula for the Cauchy integral. It is valid if L is a closed, smooth à contour enclosing the region D+ on the complex plane, and the function j z is analytic over D+, continuous over D+ L, and D- represents the region outside of L. Singular integrals H L Ü 1 j t F z ât ; z Î L 2 Π ä L t - z In this Cauchy-typeH L integral, the singular point z belongs to the contour L and under the integrand function has a H L à nonintegrable singularity. That is why this improper integral is called a singular integral. It can be evaluated, however, if a small neighborhood around this singularity t z is removed from the path of integration. The corre- sponding limit, as the size of the neighborhood shrinks to zero, is the Cauchy principal value of this divergent integral. For example: b j t x-Ε j t b j t P ât lim ât + ât ; a < x < b a t - x Ε®0 a t - x x+Ε t - x H L H L H L à à à http://functions.wolfram.com 35 In this example, P represents the Cauchy principal value, and the contour L is simply a straight segment on the real axis from a to b; in other words, L = a, b . Rather than integrating from a through the point x Î L to the point b, you can integrate on the intervals a, x - Ε and x + Ε, b and then add these results to arrive at a value. By taking Ù the limit of this calculation as Ε ® 0, you can state the principal value. H L j t j t H L H L P ât lim ât ; t0 Î L L t - t0 ·®0 L-l t - t0 This formulaH L represents HtheL Cauchy principal value of singular curvilinear integrals by the curve L with a circular à à neighborhood l, centered on t0 and of radius ·, removed. Sokhotskii formulas General 1 j t F z ât 2 Π ä L t - z F+ z zH ÎL D+ FHzL F- àz z Î D- This formula@ D represents the function F z as a piecewise analytic function in the case when L is a closed, smooth H L @ D + - contour enclosing the region D on the complex plane and D represents the region outside of L. If t0 Î L , the + - values F t0 and F t0 can be defined as the following limits: H L + + F Τ0 lim F z ; z Î D t0 Î L z®t0 H L H L - - F Τ0 lim F z ; z Î D t0 Î L H L z®t0 H L ì If L is an open contour with endpoints a and b, you can add an additional arbitrary curve segment connecting b to a H L H L ì (with the sense that F+ Τ corresponds to the limiting value from the left) and assign j Τ = 0 along this new segment. This extension allows you to apply the definitions for F+ and F- to open contours L. j t0 1 j t + H L H L F t0 + P ât ; t0 Î L 2 2 Π ä L t - t0 H L j t0 1 HjLt -H L F t0 - + Pà ât ; t0 Î L 2 2 Π ä L t - t0 These two formulasH L are calledH L the Sokhotskii formulas. They were first derived by Y. V. Sokhotskii in 1873. H L à Because they were later given a more rigorous treatment by J. Plemelj in 1908, they are often referred to as the Plemelj formulas. Sometimes the name SokhotskiiÐPlemelj formulas is used. 1 j t + - F t0 + F t0 P ât Π ä L t - t0 + - H L F t0 - F t0 j t0 H L H L à The second of these formulas can be obtained from the Sokhotskii formulas by addition and subtraction. H L H L H L http://functions.wolfram.com 36 + + - In particular, if j z is analytical over D L, then F t0 j t0 and F t0 0. If the contour L is a finite or infinite segment of the real axis, L = a, b , these formulas hold for all a < x < b, and + - so F x limΕ®H+0L F x + ä Ε , F x limÜΕ®-0 F x - äHΕ .L ThusH FL z is an HanalyticL function with a jump discontinu- ity at L, and the size of the jump is determined by the Sokhotskii formulas. H L H L H L H L H L H L Example: The exponential integral Ei 1 ¥ ã-t F z ât 2 Π ä 0 t - z After evaluation of this integral, you get: H L à ã-z Π ä - Ei z Im z > 0 F z -Π ä - Ei z Im z < 0 2 Π ä -Ei z Im z 0 H L H L ForH L arbitrary x > 0, theH LSokhotskiiH L formulas give the following values: H L H L ã-x -ã-x Ei x F+ x + 2 2 Π ä ã-x -ã-x EiH Lx F+HxL - + 2 2 Π ä 1 H¥Lã-Τ F+HxL + F- x P âΤ Π ä 0 Τ - x ã-x Ei x F+HxL + F-HxL - à Π ä F+ x - F- x ã-x H L H L H L It is important to note that for -¥ < z < 0, the function F z is analytic and its limit values taken from either side of theH realL axisH L should agree with each other. This gives the relations: lim Π ä - Ei x + ä Ε -Ei x H L Ε®0+ lim -Π ä - Ei x - ä Ε -Ei x Ε®0+ H H LL H L This leads to the following behavior of Ei x : H H LL H L lim Ei x + ä Ε Ei x + Π ä Ε®0+ H L lim Ei x - ä Ε Ei x - Π ä Ε®0+ H L H L Example: BetaH -typeL integralH L http://functions.wolfram.com 37 1 ¥ tΑ-1 F z ât 2 Π ä 0 t - z This integral can be called a beta-type integral. It can be evaluated by the following formulas: H L à 1 Π -z Α-1 csc Α Π 0 < Re Α < 1 arg z ¹ 0 F z 2 Π ä -Π zΑ-1 cot Α Π 0 < Re Α < 1 z > 0 H L H L H L ì H L ForH L arbitrary x > 0, the Sokhotskii formulas give the following results when you take into account the fact that, as Ε ® +0, -z Α-1is continuousH L for -z =H -L x +ßä Ε but has a jump of size ã-ä Α Π compared to when it is approached from the other side, -z = -x - ä Ε. H L -x - ä Ε Α-1 csc Π Α xΑ-1 ã-ä Α Π csc Α Π xΑ-1 ä F+ x lim - + cot Α Π xΑ-1 Ε®0+ 2 ä 2 ä 2 2 H-x + ä ΕLΑ-1 cscHΠ ΑL xΑ-1 ãä Α Π csc ΑH Π L xΑ-1 ä F-HxL lim - - + cotH Α ΠL xΑ-1 Ε®0+ 2 ä 2 ä 2 2 H xΑ-1L ã-ä Α ΠHcscLΑ Π xΑ-1 ãä Α Π cscH ΑL Π 1 ¥ tΑ-1 F+HxL + F- x - - ä xΑ-1 cot ΑH Π L ât 2 ä 2 ä Π ä 0 t - x xΑ-1 ãä Α Π csc Α ΠH LxΑ-1 ã-ä Α Π csc Α ΠH L F+HxL - F-HxL - xΑ-1 H L à 2 ä 2 ä H L H L IntegralH L transformsH L Exponential Fourier transform Definition 1 ¥ ä t z Ft f t z f t ã ât 2 Π -¥ This formula is the definition of the exponential Fourier transform of the function f with respect to the variable t. If @ H LD H L à H L the integral does not converge, the value of Ft f t z is defined in the sense of generalized functions for functions f t that do not grow faster than polynomials at ±¥. Properties @ H LD H L H L Linearity Ft a f t + b g t z a Ft f t z + b Ft g t z This formula reflects the linearity of the exponential Fourier transform. @ H L H LD H L H @ H LD H LL H @ H LD H LL Reflection Ft f -t z Ft f t -z @ H LD H L @ H LD H L http://functions.wolfram.com 38 This formula is called the reflection property of the exponential Fourier transform. Dilation 1 z Ft f a t z Ft f t ; a Î R a ¹ 0 a a This formula reflects the scaling or dilation property of the exponential Fourier transform. @ H LD H L @ H LD ß ¤ Shifting or translation ä a z Ft f t - a z ã Ft f t z ; a Î R This formula reflects the shifting or translation property of the exponential Fourier transform. @ H LD H L H @ H LD H LL Modulation ä a t Ft ã f t z Ft f t a + z ; a Î R This formula reflects the modulation property of the exponential Fourier transform. A H LE H L @ H LD H L ä z - b z + b Ft cos b t f a t z Ft f t - Ft f t ; a Î R b Î R 2 a a a This formula reflects the modulation property of the exponential Fourier transform. @ H L H LD H L @ H LD @ H LD ì ¤ ä z - b z + b Ft f t - Ft f t ; a Î R b Î R 2 a a a This formula reflects the modulation property of the exponential Fourier transform. @ H LD @ H LD ì ¤ Power scaling n ¶ Ft f t z n n + Ft t f t z ä ; n Î N ¶zn The power scaling Hproperty@ H LD H LshowsL that multiplication of a function by tn corresponds to the nth derivative of the @ H LD H L exponential Fourier transform. f t ¥ 1 ¥ f t ãä a t Ft z ä Ft f t z â z + ât t a 2 Π -¥ t ThisH formulaL shows that multiplication of a functionH L by t-1 corresponds to integration of the exponential Fourier B F H L à @ H LD H L à transform. Multiplication http://functions.wolfram.com 39 1 ¥ Ft f t g t z FΗ f Η z - Τ FΗ g Η Τ âΤ 2 Π -¥ The multiplication property shows that the exponential Fourier transform of a product gives the convolution of the @ H L H LD H L à I @ H LD H LM I @ H LD H LM exponential Fourier transform divided by 2 Π . Conjugation Ft f t z Ft f t -z This formula reflects the conjugation property of the exponential Fourier transform. @ H LD H L H @ H LD H LL Derivative n n k Ft f t z -ä z Ft f t z ; lim f t 0 0 £ k £ n - 1 t ®¥ H L H L The derivative property shows that the exponential Fourier transform of the nth derivative gives the product of the A H LE H L H L @ H LD H L ¤ H L í power function on the exponential Fourier transform. Integral t ä Ft f Τ âΤ z Ft f t z + g a ∆ z a z This formula shows that the exponential Fourier transform of an integral gives the product of the power function Bà H L F H L H @ H LD H LL H L H L and the exponential Fourier transform plus an expression that includes a Dirac delta function. Parseval identity ¥ ¥ f t g t ât FΤ f Τ t FΤ g Τ t ât -¥ -¥ This formula is called the Parseval identity. à H L H L à H @ H LD H LL H @ H LD H LL ¥ ¥ 2 2 f t ât FΤ f Τ t ât -¥ -¥ This Bessel’s equality follows from the Parseval identity, when g t f t . à H L¤ à @ H LD H L¤ Convolution theorem H L H L ¥ Ft f t - Τ g Τ âΤ z 2 Π Ft f t z Ft g t z -¥ This Fourier convolution theorem or convolution (Faltung) theorem for the exponential Fourier transform shows Bà H L H L F H L H @ H LD H LL H @ H LD H LL that the Fourier transform of a convolution is equal to the product of the Fourier transform multiplied by 2 Π . Relations with other integral transforms http://functions.wolfram.com 40 With inverse exponential Fourier transform -1 Ft FΤ f Τ t z f z This formula reflects the relation between direct and inverse exponential Fourier transforms. In the point z z0, whereA @f Hz LD hasH LE HaL jumpH Ldiscontinuity the composition of the inverse and direct exponential Fourier transforms 1 converges to the mean lim + f z + lim - f z . 2 z®z0 z®z0 H L With Fourier cosine and sine transforms I H L H LM f t + f -t f t - f -t Ft f t z Fct z + ä Fst z 2 2 This formula showsH L howH theL exponentialH L FourierH L transform can be represented through cosine and sine Fourier @ H LD H L B F H L B F H L transforms from even and odd parts. With Laplace transform 1 1 Ft f t z Lt f t -ä z + Lt f -t ä z 2 Π 2 Π This@ H formulaLD H L shows how@ H LtheD H exponentialL Fourier@ H LD transformH L can be represented through the Laplace transform. With Mellin transform 1 Ft f t z Mt f -log t -ä z 2 Π This@ H formulaLD H L shows how@ H the exponentialH LLD H L Fourier transform can be represented through the Mellin transform. With Z-transform ¥ ä x Ft f n ∆ t - n x Zn f n ã n=-¥ ThisB â formulaH L H showsLF H howL the@ exponentialH LD I M Fourier transform can be represented through the Z-transform. Inverse exponential Fourier transform Definition 1 ¥ -1 -ä t z Ft f t z f t ã ât 2 Π -¥ @ H LD H L à H L http://functions.wolfram.com 41 This formula is the definition of the inverse exponential Fourier transform of the function f with respect to the -1 variable t. If the integral does not converge, the value of Ft f t z is defined in the sense of generalized functions. Relations with other integral transforms @ H LD H L With exponential Fourier transform -1 Ft f t z Ft f t -z This near-equivalence identity shows that the inverse exponential Fourier transform in the point z coincides with the direct@ H LD H FourierL @ transformH LD H L in the point -z. -1 Ft f t z Ft f -t z This near-equivalence identity shows that the inverse exponential Fourier transform in the point z coincides with the direct@ H LD H FourierL @ transformH LD H L in the point -z. -1 Ft FΤ f Τ t z f z This formula reflects the relation between the direct and inverse exponential Fourier transforms. In the point z z0, whereA @f Hz L DhasH LE Ha LjumpH discontinuity,L the composition of inverse and direct exponential Fourier transforms con- 1 verges to the mean lim + f z + lim - f z . 2 z®z0 z®z0 H L Multiple exponential Fourier transform I H L H LM Definition 1 ¥ ¥ ä t1 z1+t2 z2 F t1,t2 f t1, t2 z1, z2 f t1, t2 ã ât1 ât2 2 Π -¥ -¥ H L This8 variables t1, t2. If the integral does not converge, the value of F t1,t2 f t1, t2 z1, z2 is defined in the sense of generalized functions. 8 < 1 ¥ ¥ ¥ @ H LD H L ä t z +t z +¼+tn zn F f t , t , ¼, tn z , z , ¼, zn ¼ f t , t , ¼, tn ã 1 1 2 2 ât ¼ ât ât t1,t2,¼,tn 1 2 1 2 n 1 2 n 2 1 -¥ -¥ -¥ 2 Π 2 n-timesI M 9 = This formula @is Hthe definitionLD H of the multipleL àexponentialà à FourierH transformL of the function H L n f t1, t2, ¼, tn with respect to the variables t1, t2,…, tn over R . If this integral does not converge, the value of F t1,t2,¼,tn f t1, t2, ¼, tn z1, z2, ¼, zn is defined in the sense of generalized functions. Inverse multiple exponential Fourier transform H 8 L < Definition @ H LD H L 1 ¥ ¥ F -1 f t , t z , z f t , t ã-ä t1 z1+t2 z2 ât ât t1,t2 1 2 1 2 1 2 1 2 2 Π -¥ -¥ H L 8 < @ H LD H L à à H L http://functions.wolfram.com 42 This formula is the definition of the inverse double exponential Fourier transform of the function f with respect to the variables t , t . If this integral does not converge, the value of F -1 f t , t z , z is defined in the sense of 1 2 t1,t2 1 2 1 2 generalized functions. 8 < 1 ¥ ¥ ¥ @ H LD H L -1 -ä t1 z1+t2 z2+¼+tn zn F f t1, t2, ¼, tn z1, z2, ¼, zn ¼ f t1, t2, ¼, tn ã ât ¼ ât ât t1,t2,¼,tn n n 2 1 -¥ -¥ -¥ 2 Π 2 n-timesI M 9 = This formula is@ theH definitionLD H of the inverseL multipleà à exponentialà H FourierL transform of the function H L n f t1, t2, ¼, tn with respect to the variables t1, t2,…, tn over R . If this integral does not converge, the value of F -1 f t , t , ¼, t z , z , ¼, z is defined in the sense of generalized functions. t1,t2,¼,tn 1 2 n 1 2 n Relations with other integral transforms H 8 L < @ H LD H L With multiple exponential Fourier transform -1 F f t1, t2, ¼, tn z1, z2, ¼, zn F f t1, t2, ¼, tn -z1, -z2, ¼, -zn t1,t2,¼,tn t1,t2,¼,tn This9 near= -equivalence identity shows that9 the inverse= multiple exponential Fourier transform in the point z coin- @ H LD H L @ H LD H L cides with the direct multiple exponential Fourier transform at the point -z. Fourier transform (continuous ® discrete) Definition L 2 Π ä k t 1 - FcdL;t f t k f t ã L ât ; k Î Z L 0 General properties @ H LD H L à H L Linearity FcdL;t a f t + b g t k a FcdL;t f t k + b FcdL;t g t k Reflection @ H L H LD H L @ H LD H L @ H LD H L FcdL;t f -t k FcdL;t f t -k Dilation @ H LD H L @ H LD H L Fcd f t k m k L;t m + FcdL;t f m t k ; m Î N 0 True J @ H LD J NN Shifting or @translationH LD H L 2 Π ä k a - FcdL;t f t - a k ã L FcdL;t f t k @ H LD H L @ H LD H L http://functions.wolfram.com 43 Modulation 2 Π ä m t FcdL;t ã L f t k FcdL;t f t k - m ; m Î Z MultiplicatiBon H LF H L @ H LD H L 1 ¥ FcdL;t f t g t k FcdL;t f t j FcdL;t g t k - j L j=-¥ Conjugatio@nH L H LD H L â H @ H LD H LL H @ H LD H LL FcdL;t f t k FcdL;t f t -k Derivative @ H LD H L @ H LD H L ¶ f t 2 Π ä k FcdL;t k FcdL;t f t k ¶t L H L Grouping B F H L @ H LD H L m-1 t - j L + FcdL;t f k m FcdL;t f t k m ; m Î N j=0 m SummationBâ F H L H @ H LD H LL ¥ 1 k FcdL;t f t - j L k FcdL;t f t j=-¥ L L Parseval identityB â H LF H L @ H LD L ¥ f Τ g Τ âΤ FcdL;t f t j FcdL;t g t j 0 j=-¥ Convolutionà H L theoremH L â H @ H LD H LL H @ H LD H LL L FcdL;t f Τ g t - Τ âΤ k L FcdL;t f t k FcdL;t g t k 0 Relations with other integral transforms Bà H L H L F H L H @ H LD H LL H @ H LD H LL With inverse Fourier transform (continuous ® discrete) -1 FcdL;t FcdL;k f k t k f k A @ H LD H LE H L H L http://functions.wolfram.com 44 Inverse Fourier transform (continuous ® discrete) Definition ¥ 2 Π ä k x -1 1 FcdL;k f k x f k ã L L k=-¥ Relations@ H L DwithH L other integralâ H L transforms With Fourier transform (continuous ® discrete) -1 FcdL;t FcdL;k f k t k f k Fourier transform (discrete ® continuous) A @ H LD H LE H L H L Definition ¥ 2 Π ä k x 1 - FdcL;k f k x f k ã L L k=-¥ General@ H propertiesLD H L â H L Linearity FdcL;k a f k + b g k x a FdcL;k f k x + b FdcL;k g k x Reflection @ H L H LD H L @ H LD H L @ H LD H L FdcL;k f -k x FdcL;k f k -x Dilation @ H LD H L @ H LD H L 1 m-1 x - j L + FdcL;k f k m x FdcL;k f k ; m Î N m j=0 m Shifting or @translationH LD H L â @ H LD 2 Π ä x m - FdcL;k f k - m x ã L FdcL;k f k x ; m Î Z Modulation @ H LD H L @ H LD H L 2 Π ä a k FdcL;k ã L f k x FdcL;k f k x - a Power scalingB H LF H L @ H LD H L http://functions.wolfram.com 45 L ¶ FdcL;k f k x FdcL;k k f k x - 2 Π ä ¶x H @ H LD H LL Multiplicatio@ nH LD H L 1 L FdcL;k f k g k x FdcL;k f k t FdcL;k g k x - t ât L 0 Conjugation@ H L H LD H L à H @ H LD H LL H @ H LD H LL FdcL;k f k x FdcL;k f k -x Sampling @ H LD H L @ H LD H L k ¥ FdcL;k f x FdcL;k f k x - j L L j=-¥ Zero packingB F H L â H @ H LD H LL f k m k m + FdcL;k x FdcL;k f k m x ; m Î N 0 True J N Parseval identityB F H L H @ H LD H LL ¥ L f j g j FdcL;k f k t FdcL;k g k t ât j=-¥ 0 Convolutionâ H L theoremH L à H @ H LD H LL H @ H LD H LL ¥ FdcL;k f j g k - j x L FdcL;k f k x FdcL;k g k x j=-¥ RelationsB â withH L otherH L FintegralH L transformsH @ H LD H LL H @ H LD H LL With inverse Fourier transform (discrete ® continuous) -1 FdcL;k FdcL;t f t k x f x Inverse Fourier transform (discrete ® continuous) A @ H LD H LE H L H L Definition L 2 Π ä k t -1 1 FdcL;t f t k f t ã L ât ; k Î Z L 0 @ H LD H L à H L http://functions.wolfram.com 46 Relations with other integral transforms With Fourier transform (discrete ® continuous) -1 FdcL;k FdcL;t f t k x f x Fourier transform (discrete ® discrete) A @ H LD H LE H L H L Definition m-1 2 Π ä k n 1 - m + Fddm;k f k n f k ã ; m Î N n Î Z m k=0 General@ H propertiesLD H L â H L ì Linearity + Fddm;k a f k + b g k n a Fddm;k f k n + b Fddm;k g k n ; m Î N n Î Z Reflection @ H L H LD H L @ H LD H L @ H LD H L ì + Fddm;k f -k n Fddm;k f k -n ; m Î N n Î Z Dilation @ H LD H L @ H LD H L ì j-1 + + Fddm;k f k j n Fddm;k f k n - i m ; m Î N n Î Z j Î N i=0 + Fddm;k@ f Hk jLD HnL âFddH m;k f k@ H rLDnH ; m ÎLLN n Î Zì j Î Zì gcd j, m 1 j r 1 mod m j-1 Fdd f r n-i m j n i=0 m;r j + + Fddm;k@ f Hk jLD HnL H @ H LD H LL ì ; mìÎ N ìn Î ZH Lj Î Nì j m 0 True Ú J @ H LD J NN Shifting or translation@ H LD H L í í í 2 Π ä j n - m + Fddm;k f k - j n ã Fddm;k f k n ; m Î N n Î Z j Î Z Modulation @ H LD H L @ H LD H L ì ì 2 Π ä k j m + Fddm;k ã f k n Fddm;k f k n - j ; m Î N n Î Z j Î Z MultiplicatioBn H LF H L @ H LD H L ì ì 1 m-1 + Fddm;k f k g k n Fddm;k f k j Fddm;k g k n - j ; m Î N n Î Z j Î Z m j=0 @ H L H LD H L âH @ H LD H LL H @ H LD H LL ì ì http://functions.wolfram.com 47 Conjugation Fddm;k f k n Fddm;k f k -n Repeat @ H LD H L @ H LD H L Fdd f k n j n m;k j + Fddm;k f k n ; m Î N n Î Z j Î Z 0 True @ H LD J N Zero packing@ H LD H L ì ì f k j k 1 j + Fddm;k n Fddm;k f k n ; m Î N n Î Z j Î Z 0 True j J N SummationB F H L @ H LD H L ì ì j-1 + Fddm;k f k - i m n j Fddm;k f k j n ; m Î N n Î Z j Î Z i=0 Parseval identityBâ H LF H L H @ H LD H LL ì ì m-1 m-1 + f j g j Fddm;k f k j Fddm;k g k j ; m Î N n Î Z j=0 j=0 Convolutionâ H L theoremH L â H @ H LD H LL H @ H LD H LL ì m-1 + Fddm;k f j g k - j n m Fddm;k f k n Fddm;k g k n ; m Î N n Î Z j=0 RelationsBâ withH L HotherLF integralH L transformsH @ H LD H LL H @ H LD H LL ì With inverse Fourier transform (discrete ® discrete) -1 + Fddm;k Fddm;j f j k n f n ; m Î N n Î Z Inverse Fourier transform (discrete ® discrete) A @ H LD H LE H L H L ì Definition 1 m-1 2 Π ä k n -1 m + Fddm;k f k n f k ã ; m Î N n Î Z m k=0 Relations@ H LwithD H L other integralâ H L transforms ì http://functions.wolfram.com 48 With Fourier transform (discrete ® discrete) -1 + Fddm,k Fddm;j f j k n f n ; m Î N n Î Z Fourier cosine transform A @ H LD H LE H L H L ì Definition 2 ¥ Fct f t z f t cos t z ât Π 0 This formula is the definition of the Fourier cosine transform of the function f with respect to the variable t. If the @ H LD H L à H L H L integral does not converge, the value of Fct f t z is defined in the sense of generalized functions. General properties @ H LD H L Linearity Fct a f t + b g t z a Fct f t z + b Fct g t z This formula reflects the linearity of the Fourier cosine transform. @ H L H LD H L @ H LD H L @ H LD H L Scaling 1 z Fct f a t z Fct f t ; a > 0 a a This formula reflects the scaling property of the Fourier cosine transform. @ H LD H L @ H LD Modulation 1 z + b z - b Fct cos b t f a t z Fct f t + Fct f t ; a > 0 b > 0 2 a a a This formula reflects the modulation property of the Fourier cosine transform. @ H L H LD H L @ H LD @ H LD ì 1 z + b z - b Fct sin b t f a t z Fst f t - Fst f t ; a > 0 b > 0 2 a a a This formula reflects the modulation property of the Fourier cosine transform. @ H L H LD H L @ H LD @ H LD ì Parity f t + a + f -t - a f t - a - f a - t Fct - z sin a z Fst f t z ; a Î R 2 2 This formulaH L showsH Lhow theH FourierL H cosineL transform can be applied to the difference between the even part of B F H L H L H @ H LD H LL f t + a and the odd part of f t - a . H L H L http://functions.wolfram.com 49 f t + a - f -t - a f t - a + f a - t Fct + z cos a z Fct f t z ; a Î R 2 2 This formulaH L showsH howL theH FourierL H cosineL transform can be applied to the sum of the even part of f t - a and the B F H L H L H @ H LD H LL odd part of f t + a . H L Power scaling H L 2 n ¶ Fct f t z 2 n n + Fct t f t z -1 ; n Î N ¶z2 n H @ H LD H LL 2 n th ThisA formulaH LE H L showsH Lthat multiplication of a function by t corresponds to the 2 n derivative of the Fourier cosine transform. 2 n+1 ¶ Fst f t z 2 n+1 n Fct t f t z -1 ; n Î N ¶z2 n+1 H @ H LD H LL 2 n+1 th ThisA formulaH LE HshowsL H thatL multiplication of a function by t corresponds to the 2 n + 1 derivative of the Fourier sine transform. H L Derivative 2 n-1 2 n n 2 n k 2 k 2 n-2 k-1 k + Fct f t z -1 z Fct f t z - -1 z f 0 ; lim f t 0 0 £ k £ 2 n - 1 n Î N t®¥ Π k=0 H L H L H L ThisA formulaH LE H L showsH L that the@ FourierH LD H L cosine âtransformH L of an even-orderH L derivativeH L ígives the productí of the power function with the Fourier cosine transform plus some even polynomial. 2 n-1 2 n+1 n 2 n+1 k 2 k 2 n-2 k k Fct f t z -1 z Fst f t z - -1 z f 0 ; lim f t 0 0 £ k £ 2 n n Î N t®¥ Π k=0 H L H L H L ThisA formulaH LE H showsL H thatL the Fourier@ H LD HcosineL transformâH L of an odd-orderH L derivativeH L givesí the productí of a power function with the Fourier sine transform plus some even polynomial. Convolution related ¥ Fct f Τ g t + Τ + g t - Τ âΤ z 2 Π Fct f t z Fct g t z 0 This formula shows that the Fourier cosine transform of a convolution gives the product of Fourier cosine trans- Bà H L H H L H ¤LL F H L H @ H LD H LL H @ H LD H LL forms multiplied by 2 Π . Integral ¥ 1 Fct f Τ âΤ z Fst f t z t z Bà H L F H L H @ H LD H LL http://functions.wolfram.com 50 This formula shows that the Fourier cosine transform of an indefinite integral with a variable lower limit gives the product of the Fourier sine transforms by 1 z. Limit at infinity lim Fct f t z 0 z®¥ This RiemannÐLebesgue theorem shows that the Fourier cosine transform Fct f t z converges to zero as z tends H @ H LD H LL to infinity for some classes of the function f t . Relations with other integral transforms @ H LD H L H L With inverse Fourier cosine transform -1 Fct FcΤ f Τ t z f z This formula reflects the relation between the direct and the inverse Fourier cosine transforms. In the point z z0, whereA f@ zH hasLD H L EaH jumpL Hdiscontinuity,L the composition of the inverse and the direct Fourier cosine transforms 1 converges to the mean lim + f z + limz®z- f z . 2 z®z0 0 H L With exponential Fourier transformJ H L H LN Fct f t z Fct f -t Θ -t + f t Θ t z This formula reflects the relation between the direct and the inverse exponential Fourier transforms. @ H LD H L @ H L H L H L H LD H L With Laplace transform 1 1 Fct f t z Lt f t ä z + Lt f t -ä z 2 Π 2 Π This@ HformulaLD H L represents@ theH LD HFourierL cosine @transformH LD H L through Laplace transforms. Inverse Fourier cosine transform Definition ¥ -1 2 Fct f t z f t cos t z ât Π 0 This formula is the definition of the inverse Fourier cosine transform of the function f with respect to the variable @ H LD H L à H L H L -1 t. If the integral does not converge, the value of Fct f t z is defined in the sense of generalized functions. Relations with other integral transforms @ H LD H L With Fourier cosine transform http://functions.wolfram.com 51 -1 Fct f t z Fct f t z This formula shows that the inverse Fourier cosine transform coincides with the direct Fourier cosine transform. @ H LD H L @ H LD H L Fourier sine transform Definition 2 ¥ Fst f t z f t sin t z ât Π 0 This formula is the definition of the Fourier sine transform of the function f with respect to the variable t. If the @ H LD H L à H L H L integral does not converge, the value of Fst f t z is defined in the sense of generalized functions. General properties @ H LD H L Linearity Fst a f t + b g t z a Fst f t z + b Fst g t z This formula reflects the linearity of the Fourier sine transform. @ H L H LD H L @ H LD H L @ H LD H L Scaling 1 z Fst f a t z Fst f t ; a > 0 a a This formula reflects the scaling property of the Fourier sine transform. @ H LD H L @ H LD Modulation 1 z - b z + b Fst cos b t f a t z Fst f t + Fst f t ; a > 0 b > 0 2 a a a This formula reflects the modulation property of the Fourier sine transform. @ H L H LD H L @ H LD @ H LD ì 1 z - b z + b Fst sin b t f a t z Fct f t - Fct f t ; a > 0 b > 0 2 a a a This formula reflects the modulation property of the Fourier sine transform. @ H L H LD H L @ H LD @ H LD ì Parity f t - a + f a - t f t + a - f -t - a Fst - z sin a z Fct f t z ; a Î R 2 2 This formulaH L showsH L howH theL FourierH sineL transform can be applied to the difference between the even part of B F H L H L H @ H LD H LL f t - a and the odd part of f t + a . H L H L http://functions.wolfram.com 52 f t + a + f -t - a f t - a - f a - t Fst + z cos a z Fst f t z ; a Î R 2 2 This formulaH L showsH howL theH FourierL H sineL transform can be applied to a sum of the even part of f t + a and the odd B F H L H L H @ H LD H LL part of f t - a . H L Power scaling H L 2 n ¶ Fst f t z 2 n n + Fst t f t z -1 ; n Î N ¶z2 n H @ H LD H LL 2 n th ThisA formulaH LE H L showsH L that multiplication of a function by t corresponds to the 2 n derivative of the Fourier sine transform. 2 n+1 ¶ Fct f t z 2 n+1 n+1 Fst t f t z -1 ; n Î N ¶z2 n+1 H @ H LD H LL 2 n+1 th ThisA formulaH LE HshowsL H thatL multiplication of a function by t corresponds to the 2 n + 1 derivative of the Fourier cosine transform. H L Derivative 2 n-2 2 n n 2 n k 2 k+1 2 n-2 k-2 k + Fst f t z -1 z Fst f t z + -1 z f 0 ; lim f t 0 0 £ k £ 2 n - 1 n Î N t®¥ Π k=0 H L H L H L ThisA formulaH LE H L showsH L that the@ FourierH LD H L sine transformâH L of an even-orderH L derivativeH L givesí the product ofí the power function and the Fourier sine transform plus some odd-order polynomial. 2 n-1 2 n+1 n-1 2 n+1 k 2 k+1 2 n-2 k-1 k Fst f t z -1 z Fct f t z + -1 z f 0 ; lim f t 0 0 £ k £ 2 n n Î N t®¥ Π k=0 H L H L H L ThisA formulaH LE H showsL H thatL the Fourier@ H L sineD H L transformâH of Lan odd-order derivativeH L givesH L theí product of theí power function and the Fourier cosine transform plus some odd-order polynomial. Convolution related ¥ Fst f Τ g t - Τ - g t + Τ âΤ z 2 Π Fst f t z Fct g t z 0 This formula shows that the Fourier sine transform of a convolution gives the product of the Fourier sine and the Bà H L H H ¤L H LL F H L H @ H LD H LL H @ H LD H LL Fourier cosine transforms multiplied by 2 Π . Integral ¥ 1 Fst f Τ âΤ z - Fct f t z - Fct f t 0 t z Bà H L F H L H @ H LD H L @ H LD H LL http://functions.wolfram.com 53 This formula shows that the Fourier sine transform of an indefinite integral with a variable lower limit gives the difference of the Fourier cosine transforms in z and 0 multiplied by -1 z. Limit at infinity lim Fst f t z 0 z®¥ The RiemannÐLebesgue theorem shows that the Fourier sine transform Fst f t z converges to zero as z tends to H @ H LD H LL infinity for some classes of function f t . Relations with other integral transforms @ H LD H L H L With inverse Fourier sine transform -1 Fst FsΤ f Τ t z f z This formula reflects the relation between the direct and the inverse Fourier sine transforms. At the point z z0, whereA f@zH hasLD H LaE HjumpL Hdiscontinuity,L the composition of the inverse and the direct Fourier sine transforms con- 1 verges to the mean lim + f z + limz®z- f z . 2 z®z0 0 H L With exponential FourierJ transformH L H LN Fst f t z ä Fst f -t Θ -t - f t Θ t z This formula represents the Fourier sine transform through the exponential Fourier transform. @ H LD H L @ H L H L H L H LD H L With Laplace transform ä ä Fst f t z Lt f t ä z - Lt f t -ä z 2 Π 2 Π This@ HformulaLD H L represents@ theH LD HFourierL sine transform@ H LD H throughL the Laplace transforms. Inverse Fourier sine transform Definition ¥ -1 2 Fst f t z f t sin t z ât Π 0 This formula is the definition of the inverse Fourier sine transform of the function f with respect to the variable t. If @ H LD H L à H L H L -1 the integral does not converge, the value of Fst f t z is defined in the sense of generalized functions. Relations with other integral transforms @ H LD H L With Fourier sine transform http://functions.wolfram.com 54 -1 Fst f t z Fst f t z This formula shows that the inverse Fourier sine transform coincides with the direct Fourier sine transform. @ H LD H L @ H LD H L Laplace transform Definition ¥ -t z Lt f t z f t ã ât 0 This formula is the definition of the Laplace transform of the function f with respect to the variable t. If the @ H LD H L à H L integral does not converge, the value of Lt f t z is defined in the sense of generalized functions. General properties @ H LD H L Linearity Lt a f t + b g t z a Lt f t z + b Lt g t z This formula reflects the linearity of the Laplace transform. @ H L H LD H L @ H LD H L @ H LD H L Shift -a t Lt ã f t z Lt f t z + a This shift theorem shows that the Laplace transform of a product with an exponential function gives the Laplace transformA H LinE H theL shifted@ H L Dpoint.H L Power scaling n ¶ Lt f t z Lt t f t z - ¶zn This formula showsH that@ H L DdifferentiationH LL of a Laplace transform corresponds to multiplication of the original func- @ H LD H L tion by -t. n ¶ Lt f t z n n + Lt t f t z -1 ; n Î N ¶zn This formula shows thatH @differentiationH LD H LL of a Laplace transform of order n corresponds to multiplication of the @ H LD H L H L original function by -t n. Product H L 1 Γ+ä ¥ Lt f t g t z Lt f t Τ Lt g t z - Τ âΤ ; Im Γ 0 2 Π ä Γ-ä ¥ @ H L H LD H L à H @ H LD H LL H @ H LD H LL H L http://functions.wolfram.com 55 This formula represents the Laplace transform of a product f t and g t through the contour integral along a vertical line from the corresponding product of Laplace transforms. H L H L Derivative n-1 n n n-k-1 k Lt f t z z Lt f t z - z f 0 k=0 H L H L ThisA timeH LE HdifferentiationL @ H LD HrelationL â gives theH representationL for the Laplace transform of the first derivative. n-1 n n n-k-1 k Lt f t z z Lt f t z - z f 0 k=0 H L H L th ThisA timeH LE HdifferentiationL @ H LD HrelationL â gives theH representationL of the Laplace transform of the n derivative. Integral t 1 Lt f Τ âΤ z Lt f t z 0 z This formula shows that the Laplace transform of an indefinite integral gives the product of the reciprocal function Bà H L F H L @ H LD H L of z by the Laplace transform of the function. t f Τ t - Τ n-1 1 + Lt âΤ z Lt f t z ; n Î N n 0 n - 1 ! z This formulaH L H L shows that the Laplace transform of the repeated indefinite integral Bà F H L H @ H LD H LL x t t x n-1 ¼ H f t âL t ât¼ ât f t x-t ât gives the product of the power function z-n on Laplace transform. 0 0 0 0 n-1 ! n-times H L H L Ù Ù Ù H L Ù H L Convolution t Lt f t - Τ g Τ âΤ z Lt f t z Lt g t z 0 The convolution theorem or convolution (Faltung) theorem for the Laplace transform shows that the Laplace Bà H L H L F H L H @ H LD H LL H @ H LD H LL transform of a convolution is equal to the product of Laplace transforms of the convoluted functions. Limit at infinity lim z Lt f t z lim f t z®¥ t®0+ The initial value theorem shows that limit at infinity of the Laplace transform multiplied by z is the one-sided limit H @ H LD H LL H L of the initial function at zero. Limit at zero http://functions.wolfram.com 56 lim z Lt f t z lim f t z®0 t®¥ The final value theorem shows that the limit at zero of the Laplace transform multiplied by z is the limit of the H @ H LD H LL H L initial function at infinity. Sum ¥ 1 Lt f t - L k z Lt Θ t Θ L - t f t z -z L k=-¥ 1 - ã RelationsB â H withL FotherH L integral transforms@ H L H L H LD H L With inverse Laplace transform -1 Lt LΓ;Τ f Τ t z f z This formula reflects the relation between the direct and the inverse Laplace transforms. A @ H LD H LE H L H L -1 Lt LΤ f Τ t z f z This formula reflects the relation between the direct and the inverse Laplace transforms. A @ H LD H LE H L H L With exponential Fourier transform Lt f t z 2 Π Ft Θ t f t ä z This formula shows how the Laplace transform can be represented through the exponential Fourier transform. @ H LD H L H @ H L H LD H LL With Mellin transform Lt f t z Mt Θ 1 - t f -log t z This formula shows how the Laplace transform can be represented through the Mellin transform. @ H LD H L @ H L H H LLD H L With Z-transform ¥ z Lt f n ∆ t - n z Zn f n ã n=-¥ ThisB â formulaH L H showsLF H howL the@ LaplaceH LD H L transform can be represented through the Z-transform. Inverse Laplace transform Definition 1 Γ+ä ¥ -1 t p LΓ;t f t p f t ã ât 2 Π ä Γ-ä ¥ @ H LD H L à H L http://functions.wolfram.com 57 This formula is the definition of the inverse Laplace integral transform of the function f with respect to the variable t. 1 Γ+ä ¥ -1 t p Lt f t p f t ã ât 2 Π ä Γ-ä ¥ This formula is the definition of the inverse Laplace integral transform of the function f with respect to the variable @ H LD H L à H L t. -1 k k k+1 k -1 0,0,k LΓ;t f t p lim Lt f t k®¥ k! p p H L H L This@ formulaH LD H L is the PostÐWidder form @ofH theLD inverse Laplace integral transform of the function f with respect to the variable t. Multiple Laplace transform Definition ¥ ¥ -t1 z1-t2 z2 L t1,t2 f t1, t2 z1, z2 f t1, t2 ã ât1 ât2 0 0 8 < This formula is the definition of the double Laplace transform of the function f with respect to the variables t1, t2. @ H LD H L à à H L ¥ ¥ ¥ -t z -t z -¼-tn zn L f t , t , ¼, tn z , z , ¼, zn ¼ f t , t , ¼, tn ã 1 1 2 2 ât ¼ ât ât t1,t2,¼,tn 1 2 1 2 1 2 n 2 1 0 0 0 n-times 9 = @ H LD H L à à à H L This formula is the definition of multiple Laplace transforms of the function f t1, t2, ¼, tn with n respect to the variables t1, t2,…, tn over R . Mellin transform H L Definition ¥ z-1 Mt f t z f t t ât 0 This formula is the definition of the Mellin transform of the function f with respect to the variable t. If the integral @ H LD H L à H L does not converge, the value of Mt f t z is defined in the sense of generalized functions. Usually, the integral converges in the strip Α < Re z < Β, where Α and Β depend on the function f t and can assume the values ±¥. b+z For example, M Θ t - a tb z - a ; Re z < -Re b . t @b+HzLD H L H L H L General properties A H L E H L H L H L Linearity Mt a f t + b g t z a Mt f t z + b Mt g t z This formula reflects the linearity of the Mellin transform. @ H L H LD H L @ H LD H L @ H LD H L http://functions.wolfram.com 58 Scaling -z Mt f a t z a Mt f t z ; a > 0 The operation reflects the scaling of the original variable t by a positive number a in a Mellin transform. @ H LD H L @ H LD H L Power 1 Mt f z Mt f t -z t This formula reflects the Mellin transform of the function f 1 t . B F H L @ H LD H L 1 z a Mt f t z Mt f t ; a ¹ 0 a Î R a a H L The operation provides the Mellin transform of the original variable t raised to a real power a. @ H LD H L @ H LD ß ¤ Shifting a Mt t f t z Mt f t z + a The shift theorem gives the Mellin transform of a product of the original function by some power of t. @ H LD H L @ H LD H L k a ¶ Mt f t z Mt log t f t z ¶zk @ H LD H L The@ operationH L H LD HgivesL the Mellin transform of a product of the original function by a power of log t . Derivative H L n z-k-1 k + Mt f t z 1 - z n Mt f t z - n ; lim t f t 0 0 £ k £ n - 1 n Î N t®0 H L H L This formula shows that the Mellin transform of an nth derivative gives the product of a polynomial and the Mellin A H LE H L H L @ H LD H L H L í í transform of the function. ¶ n n n + Mt t f t z -1 z Mt f t z ; n Î N ¶t n This formula shows that the Mellin transform of t ¶ f t gives the product of a power function and the Mellin B H LF H L H L @ H LD H L ¶t transform. ¶ n I M H L n + Mt t f t z 1 - z Mt f t z ; n Î N ¶t n This formula shows that the Mellin transform of ¶ t f t gives the product of the power function and the Mellin B H LF H L H L @ H LD H L ¶t transform. I M H L http://functions.wolfram.com 59 ¶n tn f t + Mt z 1 - z n Mt f t z ; n Î N ¶tn H H LL ¶n tn f t This formula shows that the Mellin transform of gives the product of a polynomial and the Mellin B F H L H L @ H LD H L ¶tn transform. H H LL n n n z-k-1 k + Mt t f t z -1 z n Mt f t z ; lim t f t 0 0 £ k £ n - 1 n Î N t®0 H L H L This formula shows that the Mellin transform of tn f n t gives the product of a polynomial and the Mellin A H LE H L H L H L @ H LD H L H L í í transform. H L H L Integral t 1 Mt f Τ âΤ z - Mt f t z + 1 0 z This formula shows that the Mellin transform of an indefinite integral gives the product of -1 z and the Mellin Bà H L F H L @ H LD H L transform in the shifted point. n t Τ3 Τ2 -1 Mt ¼ f Τ1 âΤ1 âΤ2 ¼ âΤn z Mt f t z + n 0 0 0 z n This formula shows how the Mellin transformH L of a repeated indefinite integral gives the product of a rational Bà à à H L F H L @ H LD H L function and the Mellin transform in the shiftedH L point. ¥ 1 Mt f Τ âΤ z Mt f t z + 1 t z This formula shows that the Mellin transform of the indefinite integral with a variable lower limit gives the product Bà H L F H L @ H LD H L of 1 z and the Mellin transform in the shift point. ¥ ¥ ¥ 1 Mt ¼ f Τ1 âΤ1 âΤ2 ¼ âΤn z Mt f t z + n t Τ3 Τ2 z n This formula demonstrates how the Mellin transform of a repeated indefinite integral with variable lower limits Bà à à H L F H L @ H LD H L gives the product of a rational function amd HtheL Mellin transform in the shifted point. Convolution ¥ 1 t Mt f Τ g âΤ z Mt f t z Mt g t z 0 Τ Τ The Mellin convolution theorem shows that the Mellin transform of a Mellin convolution equals the product of the Bà H L F H L H @ H LD H LL H @ H LD H LL Mellin transforms. ¥ 1 b - e a + z a b-1 c e Mt t Τ f Τ g Τ t âΤ z Mt f t Mt g t a + z ; c ¹ 0 c Î R e Î R 0 c c H L B à H L H L F H L @ H LD H @ H LD H LL ß ß ¤ http://functions.wolfram.com 60 The generalized Mellin convolution theorem shows that the Mellin transform of the generalized Mellin convolution is equal to the product of the Mellin transforms. Parseval ¥ 1 Γ+ä ¥ -s f z Τ g Τ âΤ Mt f t s Mt g t 1 - s z âs 0 2 Π ä Γ-ä ¥ This formula is called MellinÐParseval’s formula. à H L H L à H @ H LD H LL H @ H LD H LL ¥ 1 Γ+ä ¥ f Τ g Τ âΤ Mt f t s Mt g t 1 - s âs 0 2 Π ä Γ-ä ¥ This formula is called Parseval’s formula. à H L H L à H @ H LD H LL H @ H LD H LL a+1 ¥ b c 1 Γ+ä ¥ a + s + 1 s 1 s a c e 1 e - Τ f b Τ g d Τ âΤ Mt f t Mt g t - d b c âs ; c ¹ 0 c Î R d Î R 0 e c 2 Π ä Γ-ä ¥ c e This representation can be used for evaluation of the general class of integrals from products of Meijer G functions. à H L H L à @ H LD @ H LD ì ì ¤ Relations with other integral transforms With inverse Mellin transform -1 Mt MΓ;Τ f Τ t s f s This formula reflects the relation between the direct and the inverse Mellin transforms. The following theorem holds:A if@ anH LanalyticalD H LE H L functionH L f s satisfies the restriction f s < K s -2 in the strip Α < Re z < Β with some -1 constant K, then the integral MΓ;s f s t is a continuous function of the variable t and is its Mellin transform in this strip. H L H L¤ ¤ H L @ H LD H L With exponential Fourier transform -t Mt f t z 2 Π Ft f ã ä z This formula shows how the Mellin transform can be represented through the exponential Fourier transform. @ H LD H L I A I ME H LM With Fourier cosine and sine transforms f ã-t + f ãt f ã-t - f ãt Mt f t z 2 Π Fct ä z + ä 2 Π Fst ä z 2 2 This formula shows howH theL MellinH L transform can be representedH L H L through the cosine and the sine Fourier trans- @ H LD H L B F H L B F H L forms from the even and odd parts of the function. With Mellin transform http://functions.wolfram.com 61 Lt f t z Mt Θ 1 - t f -log t z This formula shows how the Laplace transform can be represented through the Mellin transform. @ H LD H L @ H L H H LLD H L Inverse Mellin transform Definition 1 Γ+ä ¥ -1 -s MΓ;s f s t f s t âs 2 Π ä Γ-ä ¥ This formula is the definition of the inverse Mellin integral transform of the function f with respect to @ H LD H L à H L -1 the variable s. If the integral does not converge, the value of MΓ;s f s t is defined in the sense of generalized functions. The condition on Γ usually has the following form: Α < Γ Re s < Β, which represents a vertical strip of convergence for the integral. Changing the vertical strip of integration Α < Re s < Β leads to a @changeH LD H Lin the original function -1 MΓ;s f s t . For example, in the case of gamma function f s G s you have H L -1 -t -1 -t MΓ;s G s t ã ; Γ > 0 and MΓ,s G s t ã - 1 ; -1 < Γ < 0. H L Multiple@ H LD H L Mellin transform H L H L @ DHefLinD iHtiLon @ H LD H L ¥ ¥ z1-1 z2-1 M t1,t2 f t1, t2 z1, z2 f t1, t2 t1 t2 ât1 ât2 0 0 This formula8 < is the definition of the double Mellin transform of the function f t1, t2 with respect to @ H LD H L à à H L the variables t1, t2. ¥ ¥ ¥ z1-1 z2-1 zn-1 M f t , t , ¼, tn z , z , ¼, zn ¼ f t , t , ¼, tn t t ¼tn ât ¼ ât ât t1,t2,¼,tn 1 2 1 2 1 2 1 2 n 2 1 0 0 0 H L n-times 9 = @ H LD H L à à à H L This formula is the definition of the multiple Mellin transform of the function f t1, t2, ¼, tn with respect to the variables t1, t2,…, tn. General properties H L Convolution ¥ f Τ t1 t2 M t1,t2 g1 g2 âΤ z1, z2 Mt f t z1 + z2 Mt1 g1 t1 z1 Mt2 g2 t2 z2 0 Τ Τ Τ The8 generalized< H L Mellin convolution theorem shows that the double Mellin transform of a Mellin generalized Bà F H L H @ H LD H LL I @ H LD H LM I @ H LD H LM convolution equals the product of the Mellin transforms in the corresponding points. n n n ¥ f Τ tk M g âΤ z , z , ¼, zn Mt f t z M g t z t1,t2,¼,tn k 1 2 k tk k k k 0 Τ k=1 Τ k=1 k=1 9 = H L Bà ä F H L @ H LD â ä @ H LD H L http://functions.wolfram.com 62 The generalized Mellin convolution theorem shows that the multiple Mellin transform of a Mellin generalized convolution equals the product of the Mellin transforms in the corresponding points. Hankel transform Definition ¥ HΝ;t f t z f t t z JΝ t z ât 0 This formula is the definition of the Hankel integral transform of the function f with respect to the variable t. If this @ H LD H L à H L H L integral does not converge, the value of HΝ;t f t z is defined in the sense of generalized functions. General properties @ H LD H L 1 HΝ;t HΝ;Τ f Τ t z f z ; Re Ν > - 2 This formula shows that the inverse Hankel integral transform coincides with the direct Hankel integral transform @ @ H LD H LD H L H L H L under the restriction Re Ν > - 1 . 2 Hilbert transform H L Definition 1 ¥ f t Ht f t x P ât ; x Î R Π -¥ t - x This formula is the definitionH L of the Hilbert transform of the function f with respect to the variable t for real x. @ H LD H L à Inverse Hilbert transform Definition 1 ¥ f t -1 Ht f t x - P ât ; x Î R Π -¥ t - x This formula is the definitionH L of the inverse Hilbert transform of the function f with respect to the variable t for @ H LD H L à real x. It coincides with the direct Hilbert transform multiplied by -1. Relations with other integral transforms With Hilbert transform -1 Ht f t x -Ht f t x Z-transform @ H LD H L @ H LD H L Definition http://functions.wolfram.com 63 ¥ -n Zn f n z f n z n=0 This@ HformulaLD H L âis theH Ldefinition of the Z-transform of the function f n with respect to the discrete variable n at the complex point z. General properties H L Linearity Zn a f n + b g n z a Zn f n z + b Zn g n z This formula reflects the linearity of the Z-transform. @ H L H LD H L H @ H LD H LL H @ H LD H LL Shifting m -m -m k Zn f n - m z z Zn f n z + z f -k z k=1 This@ HformulaLD H reflectsL Hthe @shiftingH LD H L Lpropertyâ ofH theL Z-transform. -m Zn f n - m z z Zn f n z ; f -1 f -2 ¼ f -m 0 This formula reflects the shifting property of the Z-transform. @ H LD H L H @ H LD H LL H L H L H L m-1 m m -k Zn f m + n z z Zn f n z - z f k z k=0 This@ HformulaLD H reflectsL H the@ shiftingH LD H LL propertyâ HofL the Z-transform. Scaling z n Zn a f n z Zn f n a This formula reflects the scaling property of the Z-transform. @ H LD H L @ H LD ¶ m m m + Zn n f n z -1 z Zn f n z ; m Î N ¶z This formula shows that multiplication of a function f n by nm leads to repeated differentiation of the Z-transform. @ H LD H L H L H @ H LD H LL m ¶ Zn f n z m + Zn n m f n z -z ; m Î N ¶zm H L This formula shows that multiplicationH @ H LD H LL of a function f n by n gives the product of -z m and the mth derivative AH L H LE H L H L m of the Z-transform. H L H L H L http://functions.wolfram.com 64 Product 1 z Zn f n g n z Zn f n t Zn g n â t L t t The Z-transform of a product of f n and g n is represented through a contour integral along a simple circle-type @ H L H LD H L à H @ H LD H LL @ H LD contour L encircling the origin t 0 counterclockwise. All the singular points of the function Zn f n t are located inside the contour. All the singular points of the function Z g n z are located outside the contour. H L H L n t @ H LD H L Parseval @ H LD I M ¥ 1 1 f n g n Zn f n t Zn g n â t n=0 L t t âThe HParsevalL H L àtheoremH @ followsH LD H LL from@ H theLD previous relation for z 1. The integration is performed along a simple circle-type contour L encircling the origin t 0 counterclockwise. All the singular points of the function f n t are located inside the contour. All the singular points of the function g n z are located outside Zn Zn t the contour. @ H LD H L @ H LD I M Correlation ¥ 1 n-1 + f k g k - n t Zn f n t Zn g n ât ; n Î N k=0 L t âThe HpropertyL H Lis calledà theH cross@ H LD correlationH LL @ H L propertyD of the Z-transform. Convolution ¥ + Zn f k g n - k z Zn f n z Zn g n z ; g -m 0 m Î N k=0 TheBâ convolutionH L H LtheoremF H L H for@ HtheLD HZL-LtransformH @ H LD H showsLL H thatL theì Z-transform of a convolution sum is equal to the product of the corresponding Z-transforms. Limit at infinity lim Zn f n z f 0 z®¥ The analog of the Riemann-Lebesgue theorem shows that the Z-transform Zn f n z at infinity tends to the initial @ H LD H L H L value f 0 . lim z Zn f n z f 1 ; f 0 0 @ H LD H L z®¥ H L f 1 This formula shows that the Z-transform Z f n z at infinity behaves as f 0 + . H @ H LD H LL H L H L n z H L @ H LD H L H L http://functions.wolfram.com 65 Limit at one lim z - 1 Zn f n z lim f n z®1 n®¥ This formula shows that the expression z - 1 Z f n z near point z 1 behaves as lim f n . H L H @ H LD H LL H L n n®¥ Derivative by parameter H L @ H LD H L H L ¶ f n, a ¶ Zn f n, a z Zn z ¶a ¶a This formulaH L reflectsH the@ differentiationH LD H LL by parameter property of the Z-transform. B F H L Limit by parameter Zn lim f n, a z lim Zn f n, a z a®a0 a®a0 This formula reflects the evaluation limit by parameter property of the Z-transform. B H LF H L @ H LD H L Integration by parameter b b Zn f n, t ât z Zn f n, t z ât a a This formula reflects the integration by parameter property of the Z-transform. Bà H L F H L à @ H LD H L Relations with other integral transforms With inverse Z-transforms -1 Zn Zt f t n z f z This formula reflects the relation between the direct and the inverse Z-transforms. A @ H LD H LE H L H L Inverse Z-transform Definition 1 -1 n-1 Zt f t n f t t ât 2 Π ä L This formula is the definition of the inverse Z-transform of the function f t with respect to the variable t at the @ H LD H L à H L discrete point n. The contour integral is performed along a simple circle-type contour L encircling the origin t 0 counterclockwise. H L Weber transform Definition http://functions.wolfram.com 66 ¥ WΝ;t f t z t f t JΝ t z YΝ z - YΝ t z JΝ z ât 1 This formula is the definition of the Weber integral transform of the function f with respect to the variable t. @ H LD H L à H L H H L H L H L H LL Relations with other integral transforms With inverse Weber transform -1 WΝ;t WΝ;Τ f Τ t z f z The formula shows that the composition of the direct and the inverse Weber integral transforms gives the original function@ in@ aH pointLD H LD ofH L continuity.H L Inverse Weber transform Definition ¥ t f Ν, t -1 WΝ;t f Ν, t z JΝ t z YΝ t - YΝ t z JΝ t ât 2 2 0 JΝ t + YΝ t H L This @formulaH LD H givesL à the formula for HtheH inverseL H L WeberH L integralH LL transform. H L H L Summation Finite summation n ak a1 + a2 + ¼ + an k=0 âThis formula is the definition of the finite sum. n m n-m-1 ak ak + ak+m+1 ; m £ n k=0 k=0 k=0 âThis formulaâ showsâ how a finite sum can be split into two finite sums. n n Α ak Α ak k=0 k=0 âThisH formulaL â shows that a constant factor in a summand can be taken out of the sum. n n n ak ± bj ak ± bk k=0 j=0 k=0 âThis formulaâ reflectsâ the linearity of the finite sums. http://functions.wolfram.com 67 n n n log ak log ak ; -Π < arg ak £ Π k=0 k=0 k=0 âThis formulaH L representsä the conceptâ H thatL the sum of logs is equal to the log of the product, which is correct under the given restriction. n n n Π - k=0 arg zk log zk log zk - 2 Π ä ; n Î N k=0 k=0 2 Π Ú H L âThis generalH L formulaä is correct without any restrictions. n a0 1 Π f x - t 1 1 Π 1 Π + ak cos k x + bk sin k x sin n + t ât ; ak f t cos k t ât bk f t sin k t ât 2 Π -Π 2 sin t 2 Π -Π Π -Π k=1 2 H L This âformulaH isH calledL the HDirichletLL àformula for a Fourier series . à H L H L í à H L H L I M n n-1 n 2 2 ak a2 k + a2 k+1 k=0 kf=0v fk=0v âIn this formula,â theâ sum is divided into the sums of the even and odd terms. n n-1 n-2 n 3 3 3 ak a3 k + a3 k+1 + a3 k+2 k=0 kf=0v fk=0v fk=0v âIn this formula,â theâ sum of aâk is divided into three sums with the terms a3 k, a3 k+1, and a3 k+2. n n-1 n-2 n-3 n 4 4 4 4 ak a4 k + a4 k+1 + a4 k+2 + a4 k+3 k=0 kf=0v fk=0v fk=0v fk=0v âIn this formula,â theâ sum of aâk is dividedâ into four sums with the terms a4 k, a4 k+1, a4 k+2, and a4 k+3. n- j n m-1 m ak am k+j k=0 j=0 fk=0v âIn this formula,â â the sum of ak is divided into m sums with the terms amk, amk+1,…, amk+m-2, and amk+m-1. n n n n ak bj ak bj k=0 j=0 k=0 j=0 Thisâ formulaâ describesââ the multiplication rule for finite sums. n 2 n n n n 2 2 2 ak bk ak bk - ak bj - aj bk k=1 k=1 k=1 k=1 j=k+1 Thisâ formula isâ calledâ Lagrange'sâ â identity.I M http://functions.wolfram.com 68 Infinite summation (series) ¥ n ¥ n ak lim ak "Ε,Ε>0 $∆,∆>0 "n,n>∆ ak - ak < Ε n®¥ k=0 k=0 k=0 k=0 ¥ ¥ Thisâ formula âreflects the definition of the convergentâ â infinite sums (series) k=0 ak. The sum k=0 ak converges -r ¥ k r absolutely if ak O k ; r > 1. If ak ® 0 this series can converge conditionally; for example, k=0 -1 k ¥ converges conditionally if 0 < r £ 1, and absolutely for r > 1. If limk®¥ ak ¹Ú0, the series k=0 aÚk does not converge (it is a divergent series). H L Ú H L ¥ m ¥ Ú ak ak + ak+m+1 k=0 k=0 k=0 âThis formulaâ showsâ one way to separate an arbitrary finite sum from an infinite sum. ¥ ¥ Α ak Α ak k=0 k=0 âThisH formulaL â shows that a constant factor in the summands can be taken out of the sum. ¥ ¥ ¥ ak ± bj ak ± bk k=0 j=0 k=0 âThis formulaâ reflectsâ the linearity of summation. ¥ ¥ ¥ log ak log ak ; -Π < arg ak £ Π k=0 k=0 k=0 âThis formulaH L reflectsä the statementâ HthatL the sum of the logs is equal to the log of the product, which is correct under the shown restrictions. ¥ ¥ ¥ Π - k=0 arg zk log zk log zk - 2 Π ä k=0 k=0 2 Π Ú H L âThis formulaH L is äcorrect if all sums are convergent. a2 ¥ 1 Π 1 Π 1 Π 0 2 2 2 + ak + bk f t ât ; ak f t cos k t ât bk f t sin k t ât 2 k=1 Π -Π Π -Π Π -Π Parseval'sâI lemmaM reflectsà HcompletenessL inà theH trigonometricL H L í system àcos H kL t ,HsinL k t . ¥ ¥ ¥ ak a2 k + a2 k+1 k=0 k=0 k=0 8 H L H L< âIn this formula,â theâ sum is split into the sums of even and odd terms. http://functions.wolfram.com 69 ¥ ¥ ¥ ¥ ak a3 k + a3 k+1 + a3 k+2 k=0 k=0 k=0 k=0 âIn this formula,â theâ sum ofâ ak is split into three sums with the terms a3 k, a3 k+1, and a3 k+2. ¥ ¥ ¥ ¥ ¥ ak a4 k + a4 k+1 + a4 k+2 + a4 k+3 k=0 k=0 k=0 k=0 k=0 âIn this formula,â theâ sum ofâ ak is splitâ into four sums with the terms a4 k, a4 k+1, a4 k+2, and a4 k+3. ¥ m-1 ¥ ak am k+j k=0 j=0 k=0 âIn this formula,ââ the sum of ak is split into m sums with the terms amk, amk+1,…, amk+m-2, and amk+m-1. ¥ m ¥ + ak aj+k m ; m Î N k=1 j=1 k=0 âIn this formula,ââ the sum of ak is split into m sums with the terms amk+1, amk+2,…, amk+m-1, and am k+m. ¥ ¥ ¥ ¥ ak bj ak bj k=0 j=0 k=0 j=0 Thisâ formulaâ describesââ the multiplication rule for a series. 2 ¥ ¥ ¥ ¥ ¥ 2 2 2 ak bk ak bk - ak bj - aj bk k=1 k=1 k=1 k=1 j=k+1 Thisâ formula isâ calledâ Lagrange'sâ â identity.I M Double finite summation m n n m ak,j ak,j k=0 j=0 j=0 k=0 âThisâ formulaâ reflectsâ the commutativity property of finite double sums over the rectangle 0 £ k £ m, 0 £ j £ n. n n n m m-1 ak,j bm ; bm am,j + aj,m k=0 j=0 m=0 j=0 j=0 âThisâ formulaâ shows how âto rewriteâ the double sum through a single sum. m k m m ak,j ak,j k=0 j=0 j=0 k=j âThisâ formulaâ showsâ summation over the triangle 0 £ j £ k £ m in a different order. http://functions.wolfram.com 70 k m 2 m m-2 j ak,j a2 j+k,j k=0 fj=0v j=0 k=0 âThisâ formulaâ reflectsâ summation over the triangle 0 £ j £ k £ m in a different order. 2 k-1 m 2 m m-2 j-1 e u ak,j a2 j+k+1,j k=0 fj=0v j=0 k=0 âThisâ formulaâ reflectsâ summation over the triangle 0 £ j £ k-1 £ m in a different order. 2 m m m j ak,j ak,j e u k=0 j=k j=0 k=0 âThisâ formulaâ reflectsâ summation over the triangle 0 £ k £ j £ m in a different order. m p m j p m ak,j ak,j + ak,j ; p ³ m k=0 j=k j=0 k=0 j=m+1 k=0 âThisâ formulaâ reflectsâ summationâ â over the trapezium (quadrangle) 0 £ k £ m, 0 £ k £ j £ p in a different order. m -1 m p 2 2 j+1 p m ak,j ak,j + ak,j ; p ³ m k f v m k=0 j= j=0 k=0 j= k=0 2 2 â â â â â â This fformulav reflects summationf v over the trapezium (quadrangle) 0 £ k £ m, 0 £ k £ j £ p in a different order. 2 m-1 m p 2 2 j p m e u ak,j ak,j + ak,j ; p ³ m k+1 f v m+1 k=0 j= j=0 k=0 j= k=0 2 2 â â â â â â This fformulav reflects summationf v over the trapezium (quadrangle) 0 £ k £ m, 0 £ k+1 £ j £ p in a different order. 2 j m p 2 m-1 2 p m e u ak,j ak,j + ak,j ; p ³ 2 m k=0 j=2 k j=0 kf=0v j=2 m k=0 âThisâ formula âreflectsâ summationâ â over the trapezium (quadrangle) 0 £ k £ m, 0 £ 2 k £ j £ p in a different order. j m p r m-1 r p m ak,j ak,j + ak,j ; p ³ r m k=0 j=r k j=0 kf=0v j=r m k=0 + âThisâ formulaâ reflectsâ summationâ â over the trapezium (quadrangle) 0 £ k £ m, 0 £ r k £ j £ p, r Î N in a different order. http://functions.wolfram.com 71 Double infinite summation ¥ ¥ ¥ ¥ ak,j ak,j k=0 j=0 j=0 k=0 âThisâ formulaâ reflectsâ the commutative property of infinite double sums by the quadrant 0 £ k, 0 £ j. It takes place 2 2 -r under restrictions like ak,j O k + l ; r > 1, which provide absolute convergence of this double series. ¥ ¥ ¥ m m-1 ak,j bm ; bm am,IjI+ ajM,m M k=0 j=0 m=0 j=0 j=0 âThisâ formulaâ shows how âto rewriteâ the double sum through a single sum. ¥ ¥ ¥ j ak,j ak,j-k k=0 j=0 j=0 k=0 âThisâ formulaâ showsâ how to change the order in a double sum. ¥ ¥ ¥ j ak,j ak,j-k k=0 j=0 j=0 k=0 âThisâ formulaâ showsâ how to change the order in a double sum. ¥ k ¥ ¥ ak,j ak,j k=0 j=0 j=0 k=j âThisâ formulaâ reflectsâ summation over the infinite triangle 0 £ j £ k in a different order. ¥ k+m ¥ ¥ ¥ ¥ ak,j ak,j + ak,j k=0 j=0 j=0 k=0 j=m+1 k=j-m âThisâ formulaâ reflectsâ summationâ â over the infinite trapezium 0 £ j £ k + m m > 0 in a different order. k ¥ 2 ¥ ¥ ak,j a2 j+k,j ì k=0 fj=0v j=0 k=0 âThisâ formulaâ reflectsâ summation over the infinite triangle 0 £ j £ k in a different order. 2 k-1 ¥ 2 ¥ ¥ e u ak,j a2 j+k+1,j k=0 fj=0v j=0 k=0 âThisâ formulaâ reflectsâ summation over the infinite triangle 0 £ j £ k-1 in a different order. 2 e u http://functions.wolfram.com 72 ¥ ¥ ¥ j ak,j ak,j k=0 j=k j=0 k=0 âThisâ formulaâ reflectsâ summation over the infinite triangle 0 £ k £ j in a different order. m ¥ m j ¥ m ak,j ak,j + ak,j k=0 j=k j=0 k=0 j=m+1 k=0 âThisâ formulaâ reflectsâ summationâ â over the infinite trapezium (quadrangle) 0 £ k £ m, 0 £ k £ j in a different order. m -1 m ¥ 2 2 j+1 ¥ m ak,j ak,j + ak,j k f v m k=0 j= j=0 k=0 j= k=0 2 2 â â â â â â This fformulav shows the summationf v over the infinite trapezium (quadrangle) 0 £ k £ m 0 £ k £ j in a different 2 order. m-1 í e u m ¥ 2 2 j ¥ m ak,j ak,j + ak,j k+1 f v m+1 k=0 j= j=0 k=0 j= k=0 2 2 â â â â â â This fformulav shows the summationf v over the trapezium (quadrangle) 0 £ k £ m 0 £ k+1 £ j in a different order. 2 j m ¥ 2 m-1 2 ¥ m í e u ak,j ak,j + ak,j k=0 j=2 k j=0 kf=0v j=2 m k=0 âThisâ formula âshowsâ the summationâ â over the trapezium (quadrangle) 0 £ k £ m, 0 £ 2 k £ j in a different order. j m ¥ r m-1 r ¥ m ak,j ak,j + ak,j k=0 j=r k j=0 kf=0v j=r m k=0 + âThisâ formula âshowsâ summationâ â over the trapezium (quadrangle) 0 £ k £ m, 0 £ r k £ j, r Î N in a different order. Triple infinite summation ¥ ¥ ¥ ¥ k3 k3-k1 a a k1,k2,k3 k1,k2,k3-k1-k2 k1=0 k2=0 k3=0 k3=0 k1=0 k2=0 Thisâ â formulaâ shows âhowâ to âchange the order of summation in a triple sum. Multidimensional infinite summation ¥ ¥ ¥ ¥ ¥ k4 k4-k1 k4-k1-k2 a a k1,k2,k3,k4 k1,k2,k3,k4-k1-k2-k3 k1=0 k2=0 k3=0 k4=0 k4=0 k1=0 k2=0 k3=0 â â â â â â â â http://functions.wolfram.com 73 This formula shows how to change the order of summation in multiple sums. n-2 kn- k =1k j ¥ ¥ ¥ ¥ kn kn-k1 j ¼ ak ,k ,¼,kn ¼ a n-1 1 2 k1,k2,¼,kn-1,kn- k j Ú k j=1 k1=0 k2=0 kn=0 kn=0 k1=0 k2=0 kn-1=0 Ú Thisâ â formulaâ shows how âto changeâ â the orderâ of summation in multiple sums. Products Finite products n n ak exp log ak k=0 k=0 äThis formulaâ reflectsH L the property that the product is equal to the exponent from the sum of the logarithms. n n a ãak ã k=0 k k=0 Ú äThis formula reflects the property that the product is equal to the exponent from the sum of the logarithms. Infinite products ¥ ¥ ak exp log ak k=0 k=0 äThis formulaâ reflectsH L the property that the product is equal to the exponent from the sum of the logarithms. ¥ ¥ a ãak ã k=0 k k=0 Ú äThis formula reflects the property that the product is equal to the exponent from the sum of the logarithms. Operations Limit operation lim f z F "Ε,Ε>0 $∆,∆>0 "z,0< z-a <∆ f z - F < Ε z®a ¤ K H L O H H H H L ¤ LLL http://functions.wolfram.com 74 This formula reflects the definition of a limiting value F for a function f z at the point z = a, when z approaches a in any direction (the so-called epsilon-delta definition): the formula F lim f z means that the function f z has a z®a limit value F if and only if for all Ε > 0, there exists ∆ > 0 such that f Hz L - F < Ε whenever 0 < z - a < ∆. A limiting value F does not always exist, and (if it exists) it does not always coincideH L with the value of the functionH L at the point z = a (the last value also may not exist). But in the "best situation", when all the values exist and H L ¤ ¤ coincide: F f z , and the function f z is called continuous at the point z = a. lim f z F "Ε,Ε>0 $∆,∆>0 "z,z>∆ f z - F < Ε z®¥ H L H L This formula reflects the definition of a limiting value F for a function f z at the infinite point z = ¥: the formula K H L O H H H H L ¤ LLL F lim f z means that the function f z has a limit value F if and only if for all Ε > 0, there exists ∆ > 0 such z®¥ that f z - F < Ε whenever z > ∆. A limiting value F may not alwaysH Lexist, and (if it exists) may not always coincide withH L the value of the functionH atL the point z = ¥ (the last value also may not exist). But in the "best situation", when all the values exist and coincide: F f z , and the function f z is called continuous at the point H L ¤ z = ¥. lim f z + Ε f z ; f z Î C C H L H L Ε®0 This limit shows that analytic functions are continuous functions. H L H L H L H L f z + Ε - f z lim f ¢ z ; f z Î C1 C Ε®0 Ε This limitH LdefinesH L the derivative of function f at the point z, if it exists. For analytic functions this limit exists. H L H L H L 1 n n lim -1 n-k f z + k Ε f n z ; f z Î Cn C n Î N+ Ε®0 n k Ε k=0 H L th This limitâH definesL the nH derivativeL HofL a functionH L Hf withL ì an argument z. f z f ¢ z lim lim ; lim f z lim g z 0 lim f z lim g z ¥ lim f z lim g z -¥ z®a g z z®a g¢ z z®a z®a z®a z®a z®a z®a L’Hospital’H L s ruleH appearedL in the first textbook on differential calculus, Treatise of L’Hospital, in 1696. It allows H L H L ë H L H L ë H L H L f z you toH Levaluate theH L limit of the ratio of two functions lim through the limit of the ratio of their derivatives z®a g z f ¢ z lim in the cases when lim f z and lim g z are equalH L to zero or ±¥. z®a g¢ z z®a z®a H L H L b ¥ ¥ b H L lim f t, z ât f t, z ât "Ε zH,ΕLz >0 $∆,∆>0 "b,bH>∆L f t, z ât - f t, z ât < Ε z b®¥ a a a a ¥ This formula reflects the definition of HtheL H Lconvergent integral f t, z ât at the argument z. à H L à H L à aH L à H L H L b b lim f t, z ât lim f t, z ât z®z0 a a z®z0 Ù H L à H L à H L http://functions.wolfram.com 75 This formula reflects the commutativity of the two operations definite integration and limit. ¥ ¥ lim ak z lim ak z z®z0 z®z0 k=0 k=0 This âformulaH L reflectsâ the HreorderingL of the two operations infinite summation and limit. ¥ ¥ lim ak z lim ak z z®z0 z®z0 k=0 k=0 Thisä formulaH L reflectsä theH Lreordering of the two operations infinite product and limit. Π lim f t sin k t ât 0 k®¥ -Π The RiemannÐLebesgue lemma shows that sine coefficients of the trigonometric Fourier series tend to zero as à H L H L k ® ¥. Π lim f t cos k t ât 0 k®¥ -Π The RiemannÐLebesgue lemma shows that cosine coefficients of the trigonometric Fourier series tend to zero as à H L H L k ® ¥. Relations with other functions With inverse function f f -1 z z ThisH propertyL is the definition of the inverse function f -1 z and can hold without additional restrictions on z (like zIÎ D,H whereLM D is not C) for many named functions. In these situations, f z is in most cases free of branch cuts. H L For example, sin sin-1 z z; here sin-1means f -1 with f = sin, that is, the inverse sine function (do not confuse H L this with the reciprocal function 1 sin). H L H L Some of the functionsI Hf LareM invertible: their inversions f -1 can coincide with the original f , but for other values of the parameters. For example, the inverse function for the power function za is also the power function z1 a, and the H L a log z relation z1 a z takes place only under the restriction -Π < Im £ Π. In general cases the following relation a a 1 log z takes place: z1 a exp 2 ä a Π Π - Im z. H L 2 Π a I M J N H L f -1 f z z ;z Î D I M J f J J NNvN TheH L last property for the inverse function of the direct function can be valid under special restrictions for z (where typicallyH H LL D is not C). For example, cos-1 cos z z ; 0 < Re z < Π Re z 0 Im z ³ 0 Re z Π Im z £ 0. H H LL H L ê H L ì H L ê H L ì H L http://functions.wolfram.com 76 Inequalities Algebraic inequalities n 2 n n 2 2 ak bk £ ak bk ; ak Î R bj Î R k=1 k=1 k=1 Thisâ inequalityâ is calledâ the CauchyÐìSchwarzÐBuniakowsky inequality. n 2 n n 2 2 ak bk £ ak bk k=1 k=1 k=1 Thisâ inequalityâ is called¤ â the ¤CauchyÐSchwarzÐBuniakowsky inequality. n 1 p n 1 p n 1 p p p p ak + bk £ ak + bk ; ak Î R bj Î R ak ³ 0 bj ³ 0 p > 1 k=1 k=1 k=1 ThisâH inequalityL is calledâ Minkowski'sâ inequality. ì ì ì ì n 1 p n 1 p n 1 p p p p ak + bk £ ak + bk ; p > 1 k=1 k=1 k=1 Thisâ inequality¤ is calledâ ¤Minkowski'sâ ¤inequality. n 1 p n 1 q n p q 1 1 ak bk ³ ak bk ; ak Î R bj Î R ak ³ 0 bj ³ 0 + 1 p > 1 k=1 k=1 k=1 q p Thisâ inequalityâ is calledâ Hölder's inequality.í í í í í 1 p 1 q n n n 1 1 p q ak bk ³ ak bk ; + 1 p > 1 k=1 k=1 k=1 q p Thisâ inequality¤ â is called¤ Höâlder's inequality. í 1 n 1 n 1 n ak bk £ ak bk ; ak Î R bj Î R ak £ ak+1 bk £ bk+1 1 £ k £ n - 1 n k=1 n k=1 n k=1 Thisâ inequalityâ is calledâ Chebyshev's inequality.ì ì ì ì a + b ³ a b ; a > 0 b > 0 2 This inequality is called the arithmetic-geometric inequality. ì 1 n 1 n n ak ³ ak ; ak Î R ak > 0 n k=1 k=1 â ä ì http://functions.wolfram.com 77 This inequality is called the arithmetic-geometric inequality. n n p p p p ak - bj + aj - bk - aj - ak - bj - bk ³ 0 ; 0 < p £ 2 ak Î R bj Î R k=1 j=1 âa +âb I£¡ a + b¥ ¡ ¥ ¡ ¥ ¡ ¥ M ì ì This inequality is called the triangle inequality. ¤ ¤ ¤ a - b ³ a - b This inequality is called the triangle inequality. ¤ ¤ ¤¤ Integral inequalities b 2 b b f t g t ât £ f t 2 ât g t 2 ât a a a This inequality is called the CauchyÐSchwarzÐBuniakowsky inequality. à H L H L à H L à H L b b 1 p b 1 q 1 1 f t g t ât £ f t p ât g t q ât ; + 1 p > 1 a a a q p This inequality is called Hölder's inequality. à H L H L¤ à H L¤ à H L¤ í b 1 p b 1 p b 1 p f t + g t p ât £ f t p ât + g t p ât ; p > 1 a a a This inequality is called Minkowski's inequality. à H L H L¤ à H L¤ à H L¤ n b b n n-1 ¢ fk t ât £ b - a fk t ât ; fk t ³ 0 fk t > 0 1 £ k £ n k=1 a a k=1 äThisà inequalityH L His calledL àChebyshev'sä H L inequality. H L ì H L ì a b a b £ f t ât + f -1 t ât ; f ¢ t > 0 f 0 0 b £ f a 0 0 H L This inequality is called Young's inequality. à H L à @ D H L ì H L ì H L b b a+k b f t ât £ f t g t ât £ f t ât ; k g t ât f ¢ t < 0 f t ³ 0 0 £ g t £ 1 b-k a a a This inequality is called Steffensen's inequality. à H L à H L H L à H L à H L í H L í H L í H L b b b f t 2 ât f t f t ât ¼ f t f t ât a 1 a 1 2 a 1 n b b b f t f t ât f t 2 ât ¼ f t f t ât a 2 1 a 2 a 2 n ³ 0 Ù ¼H L Ù H ¼L H L ¼ Ù H ¼L H L b b b f t f t ât f t f t ât ¼ f t 2 ât Ùa nH L 1H L aÙ n H L2 Ù a HnL H L Ù H L H L Ù H L H L Ù H L http://functions.wolfram.com 78 This inequality is called Gram's inequality. b Α f t g t ât £ f a max g t ât , a £ Α £ b ; f ¢ t > 0 f a f b ³ 0 f a ³ f b a a This inequality is called Ostrowski's inequality. à H L H L H L¤ à H L 8 < H L ì H L H L ì H L¤ H L¤ http://functions.wolfram.com 79 Copyright This document was downloaded from functions.wolfram.com, a comprehensive online compendium of formulas involving the special functions of mathematics. For a key to the notations used here, see http://functions.wolfram.com/Notations/. Please cite this document by referring to the functions.wolfram.com page from which it was downloaded, for example: http://functions.wolfram.com/Constants/E/ To refer to a particular formula, cite functions.wolfram.com followed by the citation number. e.g.: http://functions.wolfram.com/01.03.03.0001.01 This document is currently in a preliminary form. If you have comments or suggestions, please email [email protected]. © 2001-2008, Wolfram Research, Inc.