Exact formula for the sum of the squares of spherical Bessel and Neumann function of the same order

Abstract

The sum of the squares of the spherical Bessel and Neumann function of the same order (SSSBN)is the square of the modulus of the Hankel function when the argument of all function are real, and is very important in theoretical physics. However, there is no exact formula for SSSBN.Corresponding formula, which has been derived by G.N.Watson[l] is an approximate formula[!], [2] valid for Re(z) > 0 ,and it can be eo (2 k -l)!! r(v + k + !) written as J ,; (z) + N; (z) �� 2 L ( 2 ) and the error term RP satisfies JrZ k=O 2k z 2k k! f V- k + !_ 2 IR I cosvJr p! I (R(v) ,p) I 2 2P • h ?p < sm - t P cosR(vJr) (2p)! p-1 where cosh2vt = �� m! (v,m) 2m sinh2m t+ R cosht �� (2m)! P m=O Upper bound of RP in the important case when v = n + !_ , is undefined since 2 r(n+l+ m cos R(vJr) =cos VJZ' = 0 ,where R stands for the real part and m!(v, m)= ) r (n +l-m ) The same formula has been derived [l]by the method called Barne's method but the error tern is very difficult to calculate. In this contribution, we will show that an exact formula exists for SSSBN when the order of the Bessel and the Neumann function is 1 . . 2 () 2 () 2 L n (2k-l) !r (n+ k+ l) n + - ,and It can be wntten as J 1 z + N 1 z = - k 2k ( ) • 2 n+-- n+- JrZ 2 z k' r n - k + 1 Proof of the formula 2 2 k=O ' In order to show that the above formula is exact, one has to establish the identity, cosh(2n+ l)t = I r (n+l+m) 2 2111sinh2111t (1) . cosht m��of(n+ 1-m) 2m! It is an easy task to show that the equation (1) holds for n= 0 and n= 1. Now, assume that the equation ( 1) is true for n �� p. It can be easily shown that cosh(2 p + 3)t = 4 cosh(2 p + 1)t. sinh 2 t + 2 cosh(2 p + 1)- cosh(2 p -I)t (2) and hence the following formula holds. cosh(2 p + 3)t " r(p +I+ m) 2 2111+2 sinh 2111+2 t " r(p +I +m) 2 2111 sinh 2111 t p-I r(p +m) 2 2111 sinh 2111 t _ ::_:_ = :L + 2 :L - :L ----7- ---7 - - - cosh t lll=o r(p + I -m) 2m! lll=o r(p + I -m) 2m! lll=o r(p -m) 2m! =l+L...J +LJ +LJ m=l r(p-m+2) (2m-2). m=l r(p+l-m) 2m! m=l r(p-m+l) (2m-1) +P 2r(2p + 1) )22P sinh2P t r(2p) 22P sinh2p t 22(p+l) sinh2(p+l) t where p = + (p v +r(2p+1) 2p! r(2) 2 -1} 2p! It can be shown that Q = I22m sinh2m t (p +m+ 1). and P = (2p + 1)22P sinh 2P t + 22(p+I) sinh 2(p+t) t m=l (p-m+ 1) !(2m) ! Hence , cosh(2p + 3)t = f r(p +m+ 2)2m sinh2m t COSht v=O r(p + 2-m) 2(m)! Since (1) is now true for n �� p + 1 , by the mathematical induction , the equation (1) is true for all n .By Nicholson's formula[1], Jv2(z)+Nv2(z)= :2 J K0(2zsinht)cosh 2wd t (3) 0 where K0 (z) = �� Je-zcosht dt is the modified Bessel function of the second kind of the -00 zero order. Substituting for cosh(2v.t) from (1) and using we obtain n (2k-1J!r(v+k+J-) J 12(z)+ N /(z)= 2 �� 2 n+-2 n+-2 71Z L..J k 2k ( 1 ) k=O 2 Z k!r V -k + 2 from which the square of the modulus of the Hankel function follows immediately.