# A physicist's guide to explicit summation formulas involving zeros of Bessel functions and related spectral sums

## TL;DR

This paper reviews methods for explicitly computing spectral sums involving Bessel function zeros, crucial for various diffusion-related applications, emphasizing practical strategies and their limitations.

## Contribution

It provides a comprehensive pedagogic overview of analytical tools and summation formulas for spectral sums involving Bessel function zeros, with practical insights.

## Key findings

- Summarizes three main strategies for spectral sum computation.
- Highlights advantages and limitations of each method.
- Provides a collection of summation formulas from literature.

## Abstract

We summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains. Such spectral sums appear as spectral expansions of heat kernels, survival probabilities, first-passage time densities, and reaction rates in many diffusion-oriented applications. As the eigenvalues are determined by zeros of an appropriate linear combination of a Bessel function and its derivative, there are powerful analytical tools for computing such spectral sums. We discuss three main strategies: representations of meromorphic functions as sums of partial fractions, Fourier-Bessel and Dini series, and direct evaluation of the Laplace-transformed heat kernels. The major emphasis is put on a pedagogic introduction, the practical aspects of these strategies, their advantages and limitations. The review summarizes many summation formulas for spectral sums that are dispersed in the literature.