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. LeviÐCivita (1896).

Definitions of the tensor functions

For all possible values of their arguments, the discrete delta functions ∆ n and ∆ n1, n2, ¼ , Kronecker delta

functions ∆n and ∆n1,n2,¼, and signature (LeviÐCivita symbol) ¶n1, n2,¼ , nd are defined by the formulas:

1 n 0 H L H L ∆ n 0 True

∆ n1, n2, ¼ 1 ; n1 n2 ¼ 0 H L

∆ n1, n2, ¼ 0 ; Ø n1 n2 ¼ 0 H L 1 n 0 ∆n H 0 TrueL

∆ ¼ Î Î ¼ n1,n2,¼ 1 ; n1 n2 n1 Q n2 Q

∆ 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 ∆ n coincides with the Kronecker delta function ∆n. In the

case of several variables, the discrete delta function ∆ n1, n2, ¼, nm coincides with Kronecker delta function

∆n ,n ,¼,n ,0: 1 2 m H L

∆ n ∆n H L

∆ n , n , ¼, n ∆ 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 ∆ n , ∆ n1, n2, ¼ , ∆n, and ∆n1,n2,¼ have the following representations through equivalent functions:

H L H L ∆ n and ∆ n1, n2, ¼, nm ∆n and ∆n1,n2,¼,nm

∆ n ∆ n1, n2, ¼, nm ; n1 n m 1 ∆n ∆ n ∆ n ∆ ∆ ∆ H L n,0H L n1,n2 n1-n2 ∆ n ∆ ∆ ∆ H L nH L ì n1,n2,¼H,nmL n1-nm,n2-nm,¼,nm-1-nm ∆ n , n , ¼, n ∆ ∆ ∆ 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 ∆ n , ∆ n1, n2, ¼ , ∆n, and ∆n1,n2,¼ can have unit values at infinity:

∆ n and ∆ n1, n2 ∆n and ∆n1,n2

∆ ¥ 0 ∆¥ HL0 H L ∆ -¥ 0 ∆ 0 H L H L -¥ ∆ ¥, -¥ 0 ∆¥,-¥ 0 H L ∆ -¥, ¥ 0 ∆-¥,¥ 0 H L SpecificH valuesL for specialized variables H L

The tensor functions ∆ n , ∆ n1, n2, ¼ , ∆n, ∆n1,n2,¼, and ¶n1, n2,¼ , nd have the following values for some specialized variables:

H L H L ∆ n ∆n ¶n

∆ 0 1 ∆0 1 ¶0 1 ∆ 1 0 ∆ 0 ¶ 1 H L 1 1 ∆ -1 0 ∆-1 0 ¶-1 1 H L ∆ 2 0 ∆2 0 ¶2 1 ¼H L ¼ ¼ H L ∆ n 0 ; n != 0 ∆n 0 ; n != 0 ¶n 1 H L

∆ n1, n2 ∆n ,n ¶n , n H L 1 2 1 2 ∆ 0, 0 1 ∆1,1 1 ¶1, 1 0 ∆ 1, 2 0 ∆ 0 ¶ 1 H L 1,2 1, 2 ∆ 2, 1 0 ∆ 0 ¶ -1 H L 2,1 2, 1 ∆ 2, 2 0 ∆ 1 ¶ 0 H L 2,2 2, 2 ¼ ¼ ¼ H L ∆ n, 0 ∆n ∆n,0 ∆n ¶z, a -1 H L ∆ 0, n ∆n ∆0,n ∆n ¶a, z 1

H L H L http://functions.wolfram.com 3

∆ n1, n2, n3 ∆n1,n2,n3 ¶n1, n2,n3

∆ 0, 0, 0 1 ∆1,1,1 1 ¶1, 1, 2 0 ∆ H0, 0, 1 L0 ∆1,1,2 0 ¶1, 2, 3 1 ∆ 0, 1, 0 0 ∆ 0 ¶ -1 H L 1,2,1 1, 3, 2 ∆ 1, 0, 0 0 ∆ 0 ¶ 1 H L 2,1,1 2, 3, 1 ∆ 1, 1, 1 0 ∆ 1 ¶ -1 H L 1,1,1 2, 1, 3 ∆ 2, 2, 2 0 ∆ 1 ¶ 1 H L 2,2,2 3, 1, 2 ∆ -1, -1, -1 0 ∆ 1 ¶ -1 H L -1,-1,-1 3, 2, 1 AnH alyticLity H L ∆ n and ∆n are nonanalytical functions defined over C. Their possible values are 0 and 1.

m ∆ n1, n2, ¼, nm and ∆n1,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 ∆ n , ∆ n1, n2, ¼ , ∆n, ∆n1,n2,¼, and ¶n1, n2,¼ , nd do not have periodicity.

Parity and symmetry quasi-permutation symmetry H L H L

The tensor functions ∆ n , ∆ n1, n2, ¼ , ∆n, and ∆n1,n2,¼ are even functions:

∆ -n ∆ n H L H L ∆ -n1, -n2, ¼, -nm ∆ n1, n2, ¼, nm H L H L ∆-n ∆n H L H L ∆ ∆ -n1,-n2,¼,-nm n1,n2,¼,nm .

The tensor functions ∆ n1, n2, ¼ , ∆n1,n2,¼, and ¶n1, n2,¼ , nd have permutation symmetry, for example:

∆ n1, n2, ¼ ∆n1,n2,¼ ¶n1, n2,¼ H L ∆ m, n ∆ n, m ∆m,n ∆n,m ¶n1, n2,¼ , nd -¶n2, n1,¼ , nd ∆ ¼ ¼ ¼ ∆ ¶ Hn1, n2, ,Lnk, , nj, , nm n1,n2,¼,nk,¼,n j,¼,nm n1,n2,¼,nk,¼,n j,¼,nd ∆ n , n , ¼, n , ¼, n , ¼, n ; n ¹ n k ¹ j ∆ 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 ∆ n , n , ¼, n ∆ n , n , ¼ , n , n ∆ ∆ ¶ 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 ∆ n and Kronecker delta function ∆n,m have the following integral representations along the interval 0, 2 Π and unit circle z 1:

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

1 2 Π ∆ n ãä t n ât 2 Π 0

1 2 Π ä t n-m ∆nH,mL à ã ât 2 Π 0 H L 1 n-m-1 ∆n,m à z â z. 2 Π ä z 1

Transformations à ¤

The tensor functions ∆ n , ∆ n1, n2, ¼ , ∆n, ∆n1,n2,¼, and ¶n1, n2,¼ , nd satisfy various identities, for example:

∆ -n ∆ n H L H L ∆ -n1, -n2, ¼, -nm ∆ n1, n2, ¼, nm H L H L ∆-n ∆n H L H L ∆ ∆ -n1,-n2,¼,-nm n1,n2,¼,nm

∆-n,m ∆n,m - Θ n - m sgn n m ; n Î Z m Î Z

∆ n1 ∆ n2 ∆ n1, n2 H ¤ ¤L H L ß ∆ n1, n2, ¼, nm ∆ nm+1, nm+2, ¼, nm+r ∆ n1, n2, ¼, nm, nm+1, nm+2, ¼, nm+r H L H L H L ∆ ∆ ∆ 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 ¶ ¶ - ¶ ∆ 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 ∆n ,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 ∆ n , ∆ n1, n2, ¼ , ∆n, ∆n1,n2,¼, and ¶n1, n2,¼ , nd have the following complex characteristics:

∆ n1, n2, ¼, nm ∆n1,n2,¼,nm ¶n1, n2,¼ , nd H L H L ∆ ¼ ∆ ¼ ∆ ∆ ¶ ¶ 2 Abs n1, n2, , nm n1, n2, , nm n1,n2,¼,nm n1,n2,¼,nm n1, n2,¼ , nd n1, n2 H ∆ ¼L -1 ∆ ¼ ∆ -1 ∆ ¶ 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 ∆ ¼ ∆ ¼ ∆ ∆ ¶ ¶ 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, ∆ ¼ ∆ ¶ 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 ∆ H H ¼ L∆L H ¼ L ∆ I ∆M ¶ 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 ∆ n and ∆n can be provided by the following formulas:

H L http://functions.wolfram.com 5

¶∆ n 0 ¶n

¶∆nH L 0. ¶n

Fractional integro-differentiation of the tensor functions ∆ n and ∆n can be provided by the following formulas:

¶Α ∆ n n-Α ∆ n ¶nΑ G 1 - Α H L

Α -Α ¶ ∆nH L n ∆nH L . ¶nΑ G 1H- Α L

Indefinite integration H L Indefinite integration of the tensor functions ∆ n and ∆n can be provided by the following formulas:

∆ z â z ∆ z z H L

∆ â z ∆ z. à Hz L z H L

Summation à The following relations represent the sifting properties of the Kronecker and discrete delta functions:

∆k,n ak an k=-¥

¥ â ∆ k, n ak a0. k=-¥

Thereâ H existL various formulas including finite summation of signature ¶n1, n2,¼ , nd , for example:

n n n n n n n ¼ ¶ ¶ ! ¼ ¶ ∆ Τ1,Τ2,¼,Τr,Νr+1,¼,Νn Τ1,Τ2,¼,Τr,Μr+1,¼,Μn r Μr+1,¼,Μn Νk,Μk . Τ1=1 Τ2=1 Τr=1 Μr+1=1 Μr+2=1 Μn=1 k=r+1

Applicationsâ â â of the tensor functions â â â ä

The tensor functions have numerous applications throughout mathematics, number theory, analysis, and other fields. http://functions.wolfram.com 6

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