DiracDelta Notations Traditional name Dirac delta function Traditional notation d x Mathematica StandardForm notation H L DiracDelta x Primary definition@ D 14.03.02.0001.01 1 ¶ d x lim ; x Î R p ¶®0 x2 + ¶2 H L Specific values Specialized values 14.03.03.0001.01 d x 0 ; x ¹ 0 Values at fixed points H L 14.03.03.0002.01 d 0 ¥ GeneralH L characteristics Domain and analyticity d x is a non-analytical function; it is generalized function defined for x Î R. 14.03.04.0001.01 xd x R 0, ¥ H L Symmetries and periodicities H L 8 < http://functions.wolfram.com 2 Parity d x is an even generalized function. 14.03.04.0002.01 d -x d x H L Periodicity H L H L No periodicity Series representations Exponential Fourier series 14.03.06.0001.01 1 ¥ d x ãä k x ; - p < x < p 2 p k=-¥ 14.03.06.0002.01 H L â 1 1 ¥ d x + cos k x ; - p < x < p 2 p p k=1 14.03.06.0003.01 H L â H L 1 ¥ k p x k p y d 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 ¥ d x ãä t x ât 2 p -¥ 14.03.07.0002.01 H L 1 ॠä d x ãä t x q t ât - p -¥ p x 14.03.07.0003.01 H L 1 à ¥ H L d x cos x t ât p -¥ 14.03.07.0004.01 H L 2 à ¥ H L d x cos x t ât p 0 14.03.07.0005.01 ¶HdLx à 1 H ¥L - t ãä x t ât ¶x 2 p ä -¥ H L à http://functions.wolfram.com 3 14.03.07.0006.01 ¶d x 1 ¥ t ã-ä x t ât ¶x 2 p ä -¥ H L Limit representationsà 14.03.09.0001.01 1 d x lim q x + 1 q 1 - x ¶ x ¶-1 ¶®+0 2 14.03.09.0014.01 H L 2 l +H 1 !!L H lL ¤ d x lim 1 - z2 ¶®¥ 2l+1 l! 14.03.09.0002.01H L H L x I M 1 - d x lim ã ¶ ¶®0 2 ¶ ¤ 14.03.09.0003.01 H L x2 1 - d x lim ã 4 ¶ ¶®+0 2 p ¶ 14.03.09.0013.01 H L 1 ä x2 d x lim ã 2 ¶ ¶®+0 2 Πä ¶ 14.03.09.0004.01 H L 1 x d x lim sin ¶®0 p ¶ ¶ 14.03.09.0005.01 H L 1 1 d x lim sin x n + n®¥ 2 p sin x 2 2 14.03.09.0006.01 H L 1 d x lim I M ¶®0 2 ¶ cosh2 x ¶ H L 14.03.09.0007.01 d x lim log cothJa xN a®¥ 14.03.09.0008.01 H L 1 H x H L¤L d x lim Ai ¶®0 ¶ ¶ 14.03.09.0009.01 H L 1 x + 1 d x lim J 1 ¶®0 ¶ ¶ ¶ 14.03.09.0010.01 H L x2 1 - 2 x ¶ d x lim ã Ln ; n Î N ¶®0 ¶ ¶ H L http://functions.wolfram.com 4 14.03.09.0011.01 a d¢ x lim a®¥ sinh a x 14.03.09.0012.01 n H L x2 1 - 1 x n H L ¶ + d x lim ã - Hn ; n Î N ¶®+0 p ¶ ¶ ¶ H L H L Transformations Transformations and argument simplifications 14.03.16.0001.01 d x d a x ; a Î R a H L H L Identitie s¤ Functional identities 14.03.17.0001.01 n d x - xj ¢ d 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 d x1 - x0 d x2 ¡- xI0 M¥ d x1 - x0 d x2 - x1 DifHferenL tHiationL H L H L Symbolic differentiation In a distributional sense, for x Î R : 14.03.20.0001.01 d n -x -1 n d n x ; n Î N H L 14.03.20.0002.01H L n m x dH Lx H 0 L; m ÎHZL n Î Z 0 £ m < n H L 14.03.20.0003.01 n n n x d HxL -1 n! d xß ; n Î ßN H L 14.03.20.0004.01 -1 n m! xn d m HxL H L Hd 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 d m x -Hx = -L 1 m -1 k f m-k x d k x - x ; 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 d n a x a-n-1 d 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 d x â x q x Definiteà H L integrationH L For the direct function itself 14.03.21.0002.01 ¥ d t ât 1 -¥ Involving the direct function à H L In the following formulas a Î R . 14.03.21.0003.01 d 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 d¢ t - a f t ât - f ¢ a ; -¥ £ -d < a < d £ ¥ -d 14.03.21.0005.01 à ¥ ¶Hm d t L ¶nH dL x - t H L¶m+n d 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 d t z 2 p 14.03.22.0006.01 @ H LD H L ãä a x Ft d t - a x 2 p 14.03.22.0007.01 @ H LD H L ¶n d t -ä n xn Ft x ; n Î N n ¶t 2 p 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 p 2 p Inverse@ FourierH LD H L exp transforms 14.03.22.0002.01 1 -1 Ft d t z 2 p Fourier@ H LD cosH L transforms 14.03.22.0003.01 2 Fct d t z p Fourier@ H LD HsinL transforms 14.03.22.0004.01 Fst d t z 0 Laplace transforms @ H LD H L 14.03.22.0005.01 Lt d t z 1 14.03.22.0009.01 -a x Lt@dHtL-D HaL x ã q a 14.03.22.0010.01 ¶n d t - a @ H LD H L -aHx L n + Lt x ã x q a ; n Î N ¶tn H L SumB matioFnH L H L Infinite summation 14.03.23.0001.01 ¥ d x - k DiracComb x k=-¥ 14.03.23.0002.01 â H L H L ¥ 1 ¥ 2 k p x ä d x - t k ã t k=-¥ t k=-¥ 14.03.23.0003.01 â H L â ¥ 1 ¥ 2 k p x d 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 d x - k ¥ p n 2 2 p n kn cos + 2 k p 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 d x G0,0 1 - x ; x Î R x < 2 14.03.26.0002.01 H L 0,0 ß d x G0,0 x + 1 ; x Î R x > -2 RepresentationsH L throughß equivalent functions 14.03.27.0001.01 d x q¢ x ; x ¹ 0 14.03.27.0003.01 ¢ dHxL q HxL 14.03.27.0002.01 dHxL d Hx1L, x2, ¼, xn ; x1 x n 1 TheH oL reHms L ì Sokhotskii's formulas 1 1 lim ¡ä p d 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 - x (also called Green's function) of Lx ` ` fulfills the equation Lx G x - x d x - x . Then the solution to the equation Lx y x f x can be represented in the form y x f x G x - x âx. 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 yn x -1 d x - 1 - d 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 p s1 d x1 P + ä p s2 d x2 + p d x1 d 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.