Diracdelta.Pdf

DiracDelta

Notations

Traditional name

Dirac delta function

Traditional notation

∆ x

Mathematica StandardForm notation H L DiracDelta x

Primary definition@ D

14.03.02.0001.01 1 ¶ ∆ x lim ; x Î R Π ¶®0 x2 + ¶2

H L Specific values

Specialized values

14.03.03.0001.01 ∆ x 0 ; x ¹ 0

Values at fixed points H L 14.03.03.0002.01 ∆ 0 ¥

GeneralH L characteristics

Domain and analyticity

∆ x is a non-analytical function; it is generalized function defined for x Î R.

14.03.04.0001.01 x∆ x R 0, ¥ H L Symmetries and periodicities H L 8 < http://functions.wolfram.com 2

Parity

∆ x is an even generalized function.

14.03.04.0002.01 ∆ -x ∆ x H L Periodicity H L H L No periodicity

Series representations

Exponential Fourier series

14.03.06.0001.01 1 ¥ ∆ x ãä k x ; - Π < x < Π 2 Π k=-¥

14.03.06.0002.01 H L â 1 1 ¥ ∆ x + cos k x ; - Π < x < Π 2 Π Π k=1

14.03.06.0003.01 H L â H L 1 ¥ k Π x k Π y ∆ x - y sin sin ; L Î R L > 0 L k=1 L L

H L â ß Integral representations

On the real axis

Of the direct function

14.03.07.0001.01 1 ¥ ∆ x ãä t x ât 2 Π -¥

14.03.07.0002.01 H L 1 à¥ ä ∆ x ãä t x Θ t ât - Π -¥ Π x

14.03.07.0003.01 H L 1 à ¥ H L ∆ x cos x t ât Π -¥

14.03.07.0004.01 H L 2 à ¥ H L ∆ x cos x t ât Π 0

14.03.07.0005.01 ¶H∆Lx à 1 H ¥L - t ãä x t ât ¶x 2 Π ä -¥

H L à http://functions.wolfram.com 3

14.03.07.0006.01 ¶∆ x 1 ¥ t ã-ä x t ât ¶x 2 Π ä -¥

H L Limit representationsà

14.03.09.0001.01 1 ∆ x lim Θ x + 1 Θ 1 - x ¶ x ¶-1 ¶®+0 2

14.03.09.0014.01

H L 2 l +H 1 !!L H lL ¤ ∆ x lim 1 - z2 ¶®¥ 2l+1 l!

14.03.09.0002.01H L H L x I M 1 - ∆ x lim ã ¶ ¶®0 2 ¶ ¤

14.03.09.0003.01

H L x2 1 - ∆ x lim ã 4 ¶ ¶®+0 2 Π ¶

14.03.09.0013.01 H L 1 ä x2 ∆ x lim ã 2 ¶ ¶®+0 2 Πä ¶

14.03.09.0004.01 H L 1 x ∆ x lim sin ¶®0 Π ¶ ¶

14.03.09.0005.01 H L 1 1 ∆ x lim sin x n + n®¥ 2 Π sin x 2 2

14.03.09.0006.01 H L 1 ∆ x lim I M ¶®0 2 ¶ cosh2 x ¶

H L 14.03.09.0007.01 ∆ x lim log cothJa xN a®¥

14.03.09.0008.01 H L 1 H x H L¤L ∆ x lim Ai ¶®0 ¶ ¶

14.03.09.0009.01 H L 1 x + 1 ∆ x lim J 1 ¶®0 ¶ ¶ ¶

14.03.09.0010.01

H L x2 1 - 2 x ¶ ∆ x lim ã Ln ; n Î N ¶®0 ¶ ¶

H L http://functions.wolfram.com 4

14.03.09.0011.01 a ∆¢ x lim a®¥ sinh a x

14.03.09.0012.01 n H L x2 1 - 1 x n H L ¶ + ∆ x lim ã - Hn ; n Î N ¶®+0 Π ¶ ¶ ¶ H L H L Transformations

Transformations and argument simplifications

14.03.16.0001.01 ∆ x ∆ a x ; a Î R a

H L H L Identitie s¤

Functional identities

14.03.17.0001.01 n ∆ x - xj ¢ ∆ f x ; f xj 0 f xj ¹ 0 ¢ j=1 f xj I M 14.03.17.0002.01 H H LL â I M ì I M ∆ x1 - x0 ∆ x2 ¡- xI0 M¥ ∆ x1 - x0 ∆ x2 - x1

DifHferenL tHiationL H L H L

Symbolic differentiation

In a distributional sense, for x Î R :

14.03.20.0001.01 ∆ n -x -1 n ∆ n x ; n Î N

H L 14.03.20.0002.01H L n m x ∆H Lx H 0 L; m ÎHZL n Î Z 0 £ m < n

H L 14.03.20.0003.01 n n n x ∆ HxL -1 n! ∆ xß ; n Î ßN

H L 14.03.20.0004.01 -1 n m! xn ∆ m HxL H L H∆ mL-n x ; m Î Z n Î Z 0 £ n < m m - n ! H L H L 14.03.20.0005.01H L H L m H L m ß ß f x ∆ m x -HΞ = -L 1 m -1 k f m-k Ξ ∆ k x - Ξ ; m Î N k k=0 H L H L H L H L H L H L âH L H L H L http://functions.wolfram.com 5

14.03.20.0006.01 ∆ n a x a-n-1 ∆ n x ; n Î N+

H L H L IntegH raL tion H L

Indefinite integration

Involving only one direct function

14.03.21.0001.02 ∆ x â x Θ x

Definiteà H L integrationH L For the direct function itself

14.03.21.0002.01 ¥ ∆ t ât 1 -¥

Involving the direct function à H L In the following formulas a Î R .

14.03.21.0003.01 d ∆ t - a f t ât f a ; -¥ £ -d < a < d £ ¥ -d

14.03.21.0004.01 à d H L H L H L ∆¢ t - a f t ât - f ¢ a ; -¥ £ -d < a < d £ ¥ -d

14.03.21.0005.01 à ¥ ¶Hm ∆ t L ¶nH ∆L x - t H L¶m+n ∆ x ât ; m Î N+ n Î N+ m n -¥ ¶t ¶t ¶xm+n

H L H L H L Integralà transforms ì

Fourier exp transforms

14.03.22.0001.01 1 Ft ∆ t z 2 Π

14.03.22.0006.01 @ H LD H L ãä a x Ft ∆ t - a x 2 Π

14.03.22.0007.01 @ H LD H L ¶n ∆ t -ä n xn Ft x ; n Î N n ¶t 2 Π H L H L B F H L http://functions.wolfram.com 6

14.03.22.0008.01 1 x Ft DiracComb t x DiracComb 2 Π 2 Π

Inverse@ FourierH LD H L exp transforms

14.03.22.0002.01 1 -1 Ft ∆ t z 2 Π

Fourier@ H LD cosH L transforms

14.03.22.0003.01 2 Fct ∆ t z Π

Fourier@ H LD HsinL transforms

14.03.22.0004.01

Fst ∆ t z 0

Laplace transforms @ H LD H L 14.03.22.0005.01

Lt ∆ t z 1

14.03.22.0009.01 -a x Lt@∆HtL-D HaL x ã Θ a

14.03.22.0010.01 ¶n ∆ t - a @ H LD H L -aHx L n + Lt x ã x Θ a ; n Î N ¶tn

H L SumB matioFnH L H L

Infinite summation

14.03.23.0001.01 ¥ ∆ x - k DiracComb x k=-¥

14.03.23.0002.01 â H L H L ¥ 1 ¥ 2 k Π x ä ∆ x - t k ã t k=-¥ t k=-¥

14.03.23.0003.01 â H L â ¥ 1 ¥ 2 k Π x ∆ x - t k 1 + 2 cos k=-¥ t k=1 t

â H L â http://functions.wolfram.com 7

14.03.23.0004.01 ¥ ¶n ∆ x - k ¥ Π n 2 2 Π n kn cos + 2 k Π x ; n Î N+ n k=-¥ ¶x k=1 2 H L â H L â Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

14.03.26.0001.01

0,0 ∆ x G0,0 1 - x ; x Î R x < 2

14.03.26.0002.01

H L 0,0 ß ∆ x G0,0 x + 1 ; x Î R x > -2

RepresentationsH L throughß equivalent functions

14.03.27.0001.01 ∆ x Θ¢ x ; x ¹ 0

14.03.27.0003.01 ¢ ∆HxL Θ HxL

14.03.27.0002.01 ∆HxL ∆ Hx1L, x2, ¼, xn ; x1 x n 1

TheH oL reHms L ì

Sokhotskii's formulas 1 1 lim ¡ä Π ∆ x + P ¶®0 x ± ä ¶ x

Green's functionH ofL one linear differential operator ` ` Let Lx be a linear differential operator. The fundamental solution G x - Ξ (also called Green's function) of Lx ` ` fulfills the equation Lx G x - Ξ ∆ x - Ξ . Then the solution to the equation Lx y x f x can be represented in the form y x f Ξ G x - Ξ âΞ. H L H L H L H L H L The left eigenstates of the Fröbenius-Perron operator H L Ù H L H L n-1 n-1 n-1 + The functional Ψn x -1 ∆ x - 1 - ∆ x ; n Î N are the left eigenstates of the Fröbenius-Perron

operator for the r-adic map xn+1 r xn mod 1. H L H L H L I H L H LM Poincaré-Bertrand theorem http://functions.wolfram.com 8

1 1 1 1 2 lim P + ä Π s1 ∆ x1 P + ä Π s2 ∆ x2 + Π ∆ x1 ∆ x2 ; s1, s2 ±1 ¶ 1®0 x1 - ä s1 ¶ x2 - ä s2 ¶ x1 x2 ¶2®0

The generalized solutions of the linearH Ldifferential operatorsH L withH L singularH L coefficients

Linear differential operators with singular coefficients can have generalized solutions. For the hypergeometric differential equation x 1 - x y² x + Γ - Α + Β + 1 x y¢ x - Α Β y x 0 for Α, Β, Γ Î N+, Α ³ Γ > Β it can be presented in the form

Γ-Β-1 -1 k HΒ - Γ -L 1 H L H H ¥ L-1L k HΑL- Γ - 1 H L k Β+k-1 k Β+k-1 y x = c1 ∆ x + c2 ∆ x k=0 k! Β - Α - 1 k k=0 k! Α - Β - 1 k H L H L H L H L H L H L TheH L functionalâ derivative of a functionH L â H L H L H L ∆ f x The functional derivative of a function f x is ∆ x - y . ∆ f y

H L H L History H L H L

Ð O. Heaviside (1893Ð1895) Ð G. Kirchhoff (1891) Ð P. A. M. Dirac (1926) Ð L. Schwartz (1945). http://functions.wolfram.com 9

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