# $\frac{1}{\pi} \int_{0}^{2\pi} \theta^m \exp\left(-\sum_{k=1}^{K} x_k e^{-ik\pi\sin(\theta)}\right) d\theta$

## Introduction

Seeking insights on generalizing the integral solution from

$$\frac{1}{\pi} \int_{0}^{2\pi} \exp\left(-\sum_{k=1}^{K} x_k e^{-ik\pi\sin(\theta)}\right) d\theta$$Solution is here

to

$$\frac{1}{\pi} \int_{0}^{2\pi} \theta^m \exp\left(-\sum_{k=1}^{K} x_k e^{-ik\pi\sin(\theta)}\right) d\theta$$

## Questions

1. Simplification Methods: Are there techniques to simplify the analysis?
2. Solution Insights: Can the first integral's solution guide solving the second? especially for $$m = 1$$ and $$m = 2$$.
3. Literature References: Any similar forms or strategies discussed in literature?

## Context

This inquiry stems from research in wave propagation and Fourier series, where such integrals are common.

## Request

Appreciate any guidance or references. Thank you!

• I think after inserting forier series for $\theta^m$ we can solve this problem using past solution. Feb 10 at 0:13