Approximate solutions to Mathieu's equation

TL;DR

This paper reviews various approximation methods for Mathieu's equation, emphasizing their accuracy and applicability, especially in the context of Josephson junctions, to aid physicists and mathematicians working with this fundamental equation.

Contribution

It compiles and evaluates multiple approximation techniques for Mathieu's equation, providing insights into their regimes of validity and relevance to physics applications.

Findings

Certain approximations are highly accurate within specific parameter regimes

Some methods offer simple closed-form expressions for practical use

The analysis enhances understanding of Mathieu's equation in physical systems

Abstract

Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrodinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.