Integral of exp over the hypersphere

Question

I would like to know the closed form of the following integral in $\mathbb{R}^{d}$ if it exists.

It's $$ f(\boldsymbol{\xi}, a) = \int_{B(\boldsymbol{0},R)} \exp{(-i (\boldsymbol{\xi} \cdot \boldsymbol{x} - |\boldsymbol{x}|a))} d\boldsymbol{x} $$ where $a \in \mathbb{R}$ and $\boldsymbol{\xi} \in \mathbb{R}^{d}$.

In fact, when $a=0$, it has already been resolved: Integral of exp over the unit ball.

I feel this can't be written in closed form, but I appreciate your opinion.

Answer 1

Hint: In short, you have \begin{align} \int_{\mathbb{R}^d} e^{-i \xi\cdot x} f(|x|) dx \end{align} where $f(|x|)= \exp(i |x| a)\chi_{B(0, R)}$ is a radial function, i.e. Fourier transform a radial function.