Introduction
I'm grappling with an expression that intriguingly combines integrals and series, involving exponential functions with sinusoidal inputs. I'm curious about expressing this in terms of a Fourier series.
Expression
The expression in question is:
$$ \frac{\pi}{6} \int_{-\pi}^{\pi}e^{-jx\sin(\tau)}d\tau + \sum_{m \neq 0, m=-\infty}^{+\infty}\frac{1}{2\pi m^2}\int_{-\pi}^{\pi}e^{2m\tau-x\sin(\tau)}d\tau $$
Context and Motivation
My interest stems from potential applications in signal processing and the mathematical beauty of Fourier and Bessel functions, often intertwined in physical and engineering problems.
Approach and Challenges
I've attempted direct integration and series summation, confronting significant computational challenges, particularly with the series involving infinite terms and the complex nature of the integrals.
Objective
- Question: Can we represent the above expression in Fourier series terms?
- Goal: Identify methods or transformations that could simplify the expression, possibly leveraging properties of special functions.
Why It Matters
This inquiry spans several mathematical and physical disciplines, potentially enriching our toolkit for tackling wave analysis, differential equations, and beyond.
Insights, strategies, or any form of guidance are deeply appreciated.