# Uniform asymptotic formulas for the Fourier coefficients of the inverse of theta functions

## TL;DR

This paper derives uniform asymptotic formulas for the Fourier coefficients of inverse Jacobi theta functions, improving previous results and applying them to analyze the asymptotic behavior of partition rank and crank functions.

## Contribution

It introduces new uniform asymptotic formulas for Fourier coefficients of inverse theta functions and applies these to study the asymptotic properties of partition statistics.

## Key findings

- Established uniform asymptotic formulas for Fourier coefficients
- Improved upon previous results by Bringmann, Manschot, and Dousse
- Proved asymptotic monotonicity of partition rank and crank

## Abstract

In this paper, we use basic asymptotic analysis to establish some uniform asymptotic formulas for the Fourier coefficients of the inverse of Jacobi theta functions. In particular, we answer and improve some problems suggested and investigated by Bringmann, Manschot, and Dousse. As applications, we establish the asymptotic monotonicity properties for the rank and crank of the integer partitions introduced and investigated by Dyson, Andrews, and Garvan.