DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation

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1 DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition x Π lim ε ; x ε0 x 2 2 ε Specific values Specialized values x 0 ; x 0 Values at fixed points General characteristics Domain and analyticity x is a non-analytical function; it is generalized function defined for x x x 0, Symmetries and periodicities

2 2 Parity x is an even generalized function x x Periodicity No periodicity Series representations Exponential Fourier series x k x ; Π x Π k x Π cosk x ; Π x Π k 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 x t x t x Π t x Θtt x Π cosx tt Π x x 0 cosx tt x x t x t t

3 x x t x t t Limit representations Θx Θ x ε xε ε l z 2 l ε 2 l l x ε0 2 ε ε ε0 ε ε x2 4 ε Π ε x 2 2 ε ε0 Π ε sin x ε sin x n n sin x ε0 2 ε cosh 2 x ε a logcotha x ε0 ε Ai x ε ε0 ε J x ε ε x 2 2 x ε0 ε ε L n ε ; n

4 a x lim a sinha x n x lim ε Π ε x2 ε ε n H n x ε ; n Transformations Transformations and argument simplifications a x x ; a a Identities Functional identities n x x j f x ; f x j 0 f x j 0 f x j j x x 0 x 2 x 0 x x 0 x 2 x Differentiation Symbolic differentiation In a distributional sense, for x : n x n n x ; n x n m x 0 ; m n 0 m n x n n x n n x ; n x n m x n m m n mn x ; m n 0 n m m f x m x Ξ m k k 0 m k f mk Ξ k x Ξ ; m

5 n a x a n n x ; n Integration Indefinite integration Involving only one direct function x x Θx Definite integration For the direct function itself tt Involving the direct function In the following formulas a d t a f tt f a ; d a d d d d t a f tt f a ; d a d m t t m n x t t n t mn x x mn ; m n Integral transforms Fourier exp transforms t tz t t ax a x t n t n xn x ; n t n

6 t DiracCombtx DiracComb x Inverse Fourier exp transforms t tz Fourier cos transforms c t tz Fourier sin transforms s t tz 0 Laplace transforms t tz t t ax a x Θa t n t a x a x x n Θa ; n t n Summation Infinite summation x k DiracCombx k x t k 2 k Π x t t k k x t k t 2 cos 2 k Π x t k k

7 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 x G 0,0 0,0 x ; x x x G 0,0 0,0 x ; x x 2 Representations through equivalent functions x Θ x ; x x Θ x 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

8 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 ( ) G. Kirchhoff (89) P. A. M. Dirac (926) L. Schwartz (945).

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