Exact Formula for the Sum of the Squares of the Bessel Function and the Neumann Function of the Same Order of Half-Odd Integer

Abstract

Sum of the squares of the Bessel function and the Neumann function of the same order of half-odd integer has been found to be very useful in addressing a puzzle in nuclear physics. One approximate formula available in the literature is valid for the complex argument whose real part is greater than zero, and the absolute value of error term is undefined for half-odd integers. Another approximate formula which is valid for all complex arguments has been obtained using sophisticated mathematical method called Barnes' method. However, the error in the formula is very difficult to calculate. We have obtained exact formula for the sum of the squares of Bessel and Neumann functions of the same order of half-odd integers which is valid for all complex arguments, and its proof is also given.