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.

中文翻译：

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theta函数逆傅里叶系数的一致渐近公式

在本文中，我们使用基本渐近分析为Jacobi theta函数的逆傅立叶系数建立了一些统一的渐近公式。特别是，我们回答并改善了Bringmann，Manschot和Dousse提出和研究的一些问题。作为应用程序，我们为Dyson，Andrews和Garvan引入和研究的整数分区的秩和曲柄建立渐近单调性。