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General Identities 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 r f z ; m Î Z This formula reflects periodicity of function f z (if f is periodic with period r). The analytic function f z is calledH periodicL H Lif there exists a complex constant r ¹ 0 , such that f z + r f z ; z Î C. The number r (with minimal H L H L value r ) is called the period of the function f z . For example, the functions f z sin z and f z tan z have periods r 2 p and r p accordingly. H L H L r 2 p ¤ 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 f z ; the terms in a Dirichlet series include exponential functions in the form exp -lk 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 ;ICMr t - z0 r < R. 2 p ä Cr 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: 1 8 Ú q- z ¥ 1+z2 k log z = log . 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 p - arg c - Im m log z - z0 a f z r b exp 2 p ä b r + 2 p H L H H LL z H H L L r p - arg a cr - arg p - arg a c - Im m r log z - z0 p - arg c - Im m log z - z0 a cr - r + 2 p 2 p 2 p 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 p - arg c - Im m log z - z0 log a f z r 2 ä p r + 2 p H L H H LL z H H L L r p - arg a cr - arg p - arg a c - Im m r log z - z0 p - arg c - Im m log z - z0 a cr - r + + 2 p 2 p 2 p 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 p - Im log q f z p - arg a - Im r log q f z p - Im log q f z f Hz r L H L â 0 H 0L í í 0 log a q 2 ä p r + H L - Re r H L + 2 p 2 p 2 p 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 p ä r t - z0 H L TheH L Laurentâ seriesH ,L first studied àby P.
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