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Introductions to Kroneckerdelta Introduction to the Tensor Functions

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Introductions to Kroneckerdelta Introduction to the Tensor Functions Introductions to KroneckerDelta Introduction to the tensor functions General The tensor functions discrete delta and Kronecker delta first appeared in the works L. Kronecker (1866, 1903) and T. Levi–Civita (1896). Definitions of the tensor functions For all possible values of their arguments, the discrete delta functions d n and d n1, n2, ¼ , Kronecker delta functions dn and dn1,n2,¼, and signature (Levi–Civita symbol) ¶n1, n2,¼ , nd are defined by the formulas: 1 n 0 H L H L d n 0 True d n1, n2, ¼ 1 ; n1 n2 ¼ 0 H L d n1, n2, ¼ 0 ; Ø n1 n2 ¼ 0 H L 1 n 0 dn H 0 TrueL d ¼ Î Î ¼ n1,n2,¼ 1 ; n1 n2 n1 Q n2 Q d Ø ¼ n1,n2,¼ 0 ; n1 n2 . ì ì ì In other words, the Kronecker delta function is equal to 1 if all its arguments are equal. In the case of one variable, the discrete delta function d n coincides with the Kronecker delta function dn. In the case of several variables, the discrete delta function d n1, n2, ¼, nm coincides with Kronecker delta function dn ,n ,¼,n ,0: 1 2 m H L d n dn H L d n , n , ¼, n d 1 2 m n1,n2,¼,nm,0 H L t ¶n1, n2,¼ , nd -1 H L ¶n1, n2,¼ , nd 0 ; ni nj 1 £ i £ d 1 £ j £ d, H L where t is the number of permutations needed to go from the sorted version of n1, n2, ¼, nd to n1, n2, ¼, nd . ì ì Connections within the group of tensor functions and with other function groups 8 < 8 < Representations through equivalent functions http://functions.wolfram.com 2 The tensor functions d n , d n1, n2, ¼ , dn, and dn1,n2,¼ have the following representations through equivalent functions: H L H L d n and d n1, n2, ¼, nm dn and dn1,n2,¼,nm d n d n1, n2, ¼, nm ; n1 n m 1 dn d n d n d d d H L n,0H L n1,n2 n1-n2 d n d d d H L nH L ì n1,n2,¼H,nmL n1-nm,n2-nm,¼,nm-1-nm d n , n , ¼, n d d d n , n , ¼, n H L1 2 m n1,n2,¼,nm,0 n1,n2,¼,nm,0 1 2 m H L TheH best-knownL properties and formulas of theH tensor Lfunctions Simple values at zero and infinity The tensor functions d n , d n1, n2, ¼ , dn, and dn1,n2,¼ can have unit values at infinity: d n and d n1, n2 dn and dn1,n2 d ¥ 0 d¥ HL0 H L d -¥ 0 d 0 H L H L -¥ d ¥, -¥ 0 d¥,-¥ 0 H L d -¥, ¥ 0 d-¥,¥ 0 H L SpecificH valuesL for specialized variables H L The tensor functions d n , d n1, n2, ¼ , dn, dn1,n2,¼, and ¶n1, n2,¼ , nd have the following values for some specialized variables: H L H L d n dn ¶n d 0 1 d0 1 ¶0 1 d 1 0 d 0 ¶ 1 H L 1 1 d -1 0 d-1 0 ¶-1 1 H L d 2 0 d2 0 ¶2 1 ¼H L ¼ ¼ H L d n 0 ; n != 0 dn 0 ; n != 0 ¶n 1 H L d n1, n2 dn ,n ¶n , n H L 1 2 1 2 d 0, 0 1 d1,1 1 ¶1, 1 0 d 1, 2 0 d 0 ¶ 1 H L 1,2 1, 2 d 2, 1 0 d 0 ¶ -1 H L 2,1 2, 1 d 2, 2 0 d 1 ¶ 0 H L 2,2 2, 2 ¼ ¼ ¼ H L d n, 0 dn dn,0 dn ¶z, a -1 H L d 0, n dn d0,n dn ¶a, z 1 H L H L http://functions.wolfram.com 3 d n1, n2, n3 dn1,n2,n3 ¶n1, n2,n3 d 0, 0, 0 1 d1,1,1 1 ¶1, 1, 2 0 d H0, 0, 1 L0 d1,1,2 0 ¶1, 2, 3 1 d 0, 1, 0 0 d 0 ¶ -1 H L 1,2,1 1, 3, 2 d 1, 0, 0 0 d 0 ¶ 1 H L 2,1,1 2, 3, 1 d 1, 1, 1 0 d 1 ¶ -1 H L 1,1,1 2, 1, 3 d 2, 2, 2 0 d 1 ¶ 1 H L 2,2,2 3, 1, 2 d -1, -1, -1 0 d 1 ¶ -1 H L -1,-1,-1 3, 2, 1 AnH alyticLity H L d n and dn are nonanalytical functions defined over C. Their possible values are 0 and 1. m d n1, n2, ¼, nm and dn1,n2,¼,nm are nonanalytical functions defined over C . Their possible values are 0 and 1. H L ¶n1, n2,¼ , nd is a nonanalytical function, defined over the set of tuples of complex numbers with possible values 0Hand ± 1. L Periodicity The tensor functions d n , d n1, n2, ¼ , dn, dn1,n2,¼, and ¶n1, n2,¼ , nd do not have periodicity. Parity and symmetry quasi-permutation symmetry H L H L The tensor functions d n , d n1, n2, ¼ , dn, and dn1,n2,¼ are even functions: d -n d n H L H L d -n1, -n2, ¼, -nm d n1, n2, ¼, nm H L H L d-n dn H L H L d d -n1,-n2,¼,-nm n1,n2,¼,nm . The tensor functions d n1, n2, ¼ , dn1,n2,¼, and ¶n1, n2,¼ , nd have permutation symmetry, for example: d n1, n2, ¼ dn1,n2,¼ ¶n1, n2,¼ H L d m, n d n, m dm,n dn,m ¶n1, n2,¼ , nd -¶n2, n1,¼ , nd d ¼ ¼ ¼ d ¶ Hn1, n2, ,Lnk, , nj, , nm n1,n2,¼,nk,¼,n j,¼,nm n1,n2,¼,nk,¼,n j,¼,nd d n , n , ¼, n , ¼, n , ¼, n ; n ¹ n k ¹ j d ¹ ¹ -¶ H L1 2 H Lj k m k j n1,n2,¼,n j,¼,nk,¼,nm ; nk nj k j n1,n2,¼,n j,¼,nk,¼,nd d n , n , ¼, n d n , n , ¼ , n , n d d ¶ I 1 2 m 2 3 M d 1 n1, n2,¼ , nd n2, n3,¼ , nd, n1 n1, n2,¼ , nd - d+1 ¶ I M ì ì 1 n2, n3,¼ , nd, n1 H L H L Integral representations H L The discrete delta function d n and Kronecker delta function dn,m have the following integral representations along the interval 0, 2 p and unit circle z 1: H L H L ¤ http://functions.wolfram.com 4 1 2 p d n ãä t n ât 2 p 0 1 2 p ä t n-m dnH,mL à ã ât 2 p 0 H L 1 n-m-1 dn,m à z â z. 2 p ä z 1 Transformations à ¤ The tensor functions d n , d n1, n2, ¼ , dn, dn1,n2,¼, and ¶n1, n2,¼ , nd satisfy various identities, for example: d -n d n H L H L d -n1, -n2, ¼, -nm d n1, n2, ¼, nm H L H L d-n dn H L H L d d -n1,-n2,¼,-nm n1,n2,¼,nm d-n,m dn,m - q n - m sgn n m ; n Î Z m Î Z d n1 d n2 d n1, n2 H ¤ ¤L H L ß d n1, n2, ¼, nm d nm+1, nm+2, ¼, nm+r d n1, n2, ¼, nm, nm+1, nm+2, ¼, nm+r H L H L H L d d d ³ ³ n1,n2,¼,nm nm+1,nm+2,¼,nm+r n1,n2,¼,nm,nm+1,nm+2,¼,nm+r ; m 2 r 2 H L H L H L d ¶ ¶ - ¶ d n1, n2,¼ , nd m1, m2,¼ , md m1, m2,¼ , md ß nk,mk permutations m1,m2,¼,md k=1 âI M d - r ! r! ä d ¶n ,n ,¼,n ,n ,n ,¼,n ¶n ,n ,¼,n ,m ,m ,¼,m - ¶m ,m ,¼,m dn ,m . 1 2 r-1 r r+1 d 1 2 r-1 r r+1 d d! r r+1 d k k permutations mr,mr+1,¼,md k=r H L Complex characteristics âI M ä The tensor functions d n , d n1, n2, ¼ , dn, dn1,n2,¼, and ¶n1, n2,¼ , nd have the following complex characteristics: d n1, n2, ¼, nm dn1,n2,¼,nm ¶n1, n2,¼ , nd H L H L d ¼ d ¼ d d ¶ ¶ 2 Abs n1, n2, , nm n1, n2, , nm n1,n2,¼,nm n1,n2,¼,nm n1, n2,¼ , nd n1, n2 H L -1 -1 d ¼ d ¼ d d ¶ Arg Arg n1, n2, , nm tan n1, n2, , nm , 0 Arg n1,n2,¼,nm tan n1,n2,¼,nm , 0 Arg n1, n2,¼ , nd tan d ¼ d ¼ d d ¶ ¶ Re ReH n1, n2, ,L¤nm H n1, n2, ,Lnm ¡Re n1,n2,¼¥ ,nm n1,n2,¼,nm ¡Re n1, n2,¼¥, nd n1, d ¼ d ¶ Im Im H nH1, n2, , nm LL 0 H H L L Im I n1,n2,¼,nm M 0 I M Im In1, n2,¼ , nd M 0 d H H ¼ LdL H ¼ L d I dM ¶ I ¶M Conjugate n1, n2, , nm n1, n2, , nm n1,n2,¼,nm n1,n2,¼,nm n1, n2,¼ , nd n1, n2,¼ , H H LL I M I M Differentiation H L H L Differentiation of the tensor functions d n and dn can be provided by the following formulas: H L http://functions.wolfram.com 5 ¶d n 0 ¶n ¶dnH L 0. ¶n Fractional integro-differentiation of the tensor functions d n and dn can be provided by the following formulas: ¶a d n n-a d n ¶na G 1 - a H L a -a ¶ dnH L n dnH L .
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