DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values Specialized values 4.03.03.000.0 x 0 ; x 0 Values at fixed points 4.03.03.0002.0 0 General characteristics Domain and analyticity x is a non-analytical function; it is generalized function defined for x. 4.03.04.000.0 x x 0, Symmetries and periodicities
http://functions.wolfram.com 2 Parity x is an even generalized function. 4.03.04.0002.0 x x Periodicity No periodicity Series representations Exponential Fourier series 4.03.06.000.0 x k x ; Π x Π k 4.03.06.0002.0 x Π cosk x ; Π x Π k 4.03.06.0003.0 x y L sin k Π x L k sin k Π y L ; L L 0 Integral representations On the real axis Of the direct function 4.03.07.000.0 x t x t 4.03.07.0002.0 x Π t x Θtt 4.03.07.0003.0 x Π cosx tt Π x 4.03.07.0004.0 x 0 cosx tt 4.03.07.0005.0 x x t x t t
http://functions.wolfram.com 3 4.03.07.0006.0 x x t x t t Limit representations 4.03.09.000.0 Θx Θ x ε xε ε0 2 4.03.09.004.0 2l z 2 l ε 2 l l 4.03.09.0002.0 x ε0 2 ε ε ε0 ε0 4.03.09.0003.0 ε x2 4 ε 4.03.09.003.0 2Π ε x 2 2 ε 4.03.09.0004.0 ε0 Π ε sin x ε 4.03.09.0005.0 sin x n n sin x 2 2 4.03.09.0006.0 ε0 2 ε cosh 2 x ε 4.03.09.0007.0 a logcotha x 4.03.09.0008.0 ε0 ε Ai x ε 4.03.09.0009.0 ε0 ε J x ε ε 4.03.09.000.0 x 2 2 x ε0 ε ε L n ε ; n
http://functions.wolfram.com 4 4.03.09.00.0 a x lim a sinha x n x lim ε0 4.03.09.002.0 Π ε x2 ε ε n H n x ε ; n Transformations Transformations and argument simplifications 4.03.6.000.0 a x x ; a a Identities Functional identities 4.03.7.000.0 n x x j f x ; f x j 0 f x j 0 f x j j 4.03.7.0002.0 x x 0 x 2 x 0 x x 0 x 2 x Differentiation Symbolic differentiation In a distributional sense, for x : 4.03.20.000.0 n x n n x ; n 4.03.20.0002.0 x n m x 0 ; m n 0 m n 4.03.20.0003.0 x n n x n n x ; n 4.03.20.0004.0 x n m x n m m n mn x ; m n 0 n m 4.03.20.0005.0 m f x m x Ξ m k k 0 m k f mk Ξ k x Ξ ; m
http://functions.wolfram.com 5 4.03.20.0006.0 n a x a n n x ; n Integration Indefinite integration Involving only one direct function 4.03.2.000.02 x x Θx Definite integration For the direct function itself 4.03.2.0002.0 tt Involving the direct function In the following formulas a. 4.03.2.0003.0 d t a f tt f a ; d a d d 4.03.2.0004.0 d d t a f tt f a ; d a d m t t m 4.03.2.0005.0 n x t t n t mn x x mn ; m n Integral transforms Fourier exp transforms 4.03.22.000.0 t tz 4.03.22.0006.0 t t ax a x 4.03.22.0007.0 t n t n xn x ; n t n
http://functions.wolfram.com 6 4.03.22.0008.0 t DiracCombtx DiracComb x Inverse Fourier exp transforms 4.03.22.0002.0 t tz Fourier cos transforms 4.03.22.0003.0 c t tz Fourier sin transforms 4.03.22.0004.0 s t tz 0 Laplace transforms 4.03.22.0005.0 t tz 4.03.22.0009.0 t t ax a x Θa 4.03.22.000.0 t n t a x a x x n Θa ; n t n Summation Infinite summation 4.03.23.000.0 x k DiracCombx k 4.03.23.0002.0 x t k 2 k Π x t t k k 4.03.23.0003.0 x t k t 2 cos 2 k Π x t k k
http://functions.wolfram.com 7 4.03.23.0004.0 n x k 2 n k n cos Π n 2 k Π x ; n k x n 2 k Representations through more general functions Through Meijer G Classical cases for the direct function itself 4.03.26.000.0 x G 0,0 0,0 x ; x x 2 4.03.26.0002.0 x G 0,0 0,0 x ; x x 2 Representations through equivalent functions 4.03.27.000.0 x Θ x ; x 0 4.03.27.0003.0 x Θ x 4.03.27.0002.0 x x, x 2,, x n ; x x n Theorems Sokhotskii's formulas lim ε0 x ± ε Π x x Green's function of one linear differential operator Let L x be a linear differential operator. The fundamental solution Gx Ξ (also called Green's function) of L x fulfills the equation L xgx Ξ x Ξ. Then the solution to the equation L xyx f x can be represented in the form yx f ΞGx ΞΞ. The left eigenstates of the Fröbenius-Perron operator The functional Ψ n x n n x n x ; n are the left eigenstates of the Fröbenius-Perron operator for the r-adic map x n r x n mod. Poincaré-Bertrand theorem
http://functions.wolfram.com 8 lim ε 0 x s ε x 2 s 2 ε x Π s x x 2 Π s 2 x 2 Π 2 x x 2 ; s, s 2 ± ε 2 0 The generalized solutions of the linear differential operators with singular coefficients Linear differential operators with singular coefficients can have generalized solutions. For the hypergeometric differential equation x x y x Γ Α Β xy x Α Β yx 0 for Α, Β, Γ, Α Γ Β it can be presented in the form ΓΒ k Β Γ k k Α Γ yx c Βk k x c 2 Βk x k 0 kβ Α k k 0 kα Β k The functional derivative of a function The functional derivative of a function f x is f x x y. f y History O. Heaviside (893 895) G. Kirchhoff (89) P. A. M. Dirac (926) L. Schwartz (945).
http://functions.wolfram.com 9 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/0.03.03.000.0 This document is currently in a preliminary form. If you have comments or suggestions, please email comments@functions.wolfram.com. 200-2008, Wolfram Research, Inc.