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

Background and Motivation

I am exploring an integral that emerges within the context of MIMO (Multiple Input Multiple Output) communication engineering, particularly as it relates to computing conditional probability density functions (PDFs) for complex-normal (CN) distributions. The integral in question is formulated as follows:

$$f(\vec{x}) = \frac{1}{\pi} \int_0^{2\pi} \exp\left(-\sum_{k=1}^K x_k \exp(-i k \pi \sin(\theta))\right) d\theta,$$

where $$\vec{x}$$ represents a vector in a $$K$$-dimensional space.

Relevant Definitions and Previous Discussions

This inquiry extends the analysis from previously discussed cases with two variables to a more general scenario involving $$K$$ variables. Here are links to the discussions that lay the groundwork for this question:

Assuming the integral is both positive and real—as a probability density function should be—I am interested in understanding its properties and identifying techniques for its evaluation.

Current Progress and Possible Strategies

To simplify the integral, I have considered employing multinomial expansions and Fourier series, particularly leveraging the expansion of the $$J_0$$ function in terms of $$\sin(\theta)$$. The possibility of a Fourier–Bessel series representation has also arisen during my exploration.

Inquiry and Request for Insights

This integral is notably relevant in the study of quantum channels for computing conditional PDFs. I am seeking insights, references, or methodologies that could facilitate the handling of such integrals in the domain of MIMO communication or related fields.

Any guidance or suggestions on how to approach the evaluation of this integral, including transformation techniques or numerical methods suited for this type of problem, would be highly appreciated.

Bessel Functions:

Bessel functions, denoted as ( J_n(x) ), are solutions to Bessel's differential equation, which commonly arise in problems involving wave propagation, heat conduction, and other phenomena with circular or cylindrical symmetry. They are named after the German mathematician Friedrich Bessel. Bessel functions exhibit oscillatory behavior and are widely used in various branches of physics and engineering.

InHansen Bessel Formula:

The InHansen Bessel formula is a specialized form of Bessel function developed to solve certain types of engineering problems, particularly in the field of electromagnetic theory and antenna design. It's an extension of the standard Bessel functions, tailored to handle specific boundary conditions or configurations encountered in practical applications.

Update

This is normalization factor of multinomial generalized vonmises fisher distribution. It seems there is no closed-form formula for that.

• Please use MathJax to format equations.
– Gary
Commented Feb 8 at 14:57
• There is a related expression, $(16)$, for $\frac1{2\pi}\int_{-\pi}^\pi \exp\left(i\sum_\limits{k=1}^K x_k \sin(p_k\theta)\right) d \theta$ Commented Feb 8 at 15:18

You can approach this integral via Taylor series and multinomial expansion: \begin{aligned} \frac{1}{\pi}\int_0^{2\pi}\exp\left(-\sum_{k=1}^Kx_ke^{-i\pi k\sin\theta}\right)\text d\theta&=\frac{1}{\pi}\int_0^{2\pi}\sum_{n=0}^\infty\frac{(-1)^n}{n!}\left(\sum_{k=1}^Kx_ke^{-i\pi k\sin\theta}\right)^n\text d\theta\\ &=\frac{1}{\pi}\sum_{n=0}^\infty\frac{(-1)^n}{n!}\int_0^{2\pi}\sum_{r_1+r_2+...+r_K=n}\frac{n!}{\prod_{i=1}^Kr_i!}\prod_{k=1}^Kx_k^{r_k}e^{-i\pi kr_k\sin\theta}\text d\theta\\ &=\frac{1}{\pi}\sum_{n=0}^\infty\sum_{r_1+r_2+...+r_K=n}(-1)^n\prod_{k=1}^K\frac{x_k^{r_k}}{r_k!}\int_0^{2\pi}e^{-i\pi \left(\sum_{k=1}^Kkr_k\right)\sin\theta}\text d\theta\\ &=2\sum_{n=0}^\infty\sum_{r_1+r_2+...+r_K=n}(-1)^nJ_0\left(\pi\sum_{k=1}^Kkr_k\right)\prod_{k=1}^K\frac{x_k^{r_k}}{r_k!}, \end{aligned} where $$J_0(...)$$ is the Bessel function of the first kind.