Integral of the exponential in the n dimensional ball

Question

I would like to calculate the integral $\int_{||\boldsymbol{x}||_{2}\leq \epsilon} e^{\boldsymbol{c}^{T}\boldsymbol{x}}d\boldsymbol{x}$ where $\boldsymbol{x}\in\mathbb{R}^n$, $\epsilon>0$, $\boldsymbol{c}\in \mathbb{R}^n$.

Answer 1

Everything is rotationally symmetric, so you might as well take $\boldsymbol{c}$ along the positive $x$ axis and let its magnitude be $c$. Then at each given $x$ the cross section is an $n-1$ dimensional ball of radius $\sqrt{2^2-x^2}$ Now your integral becomes $\int_{-2}^2e^{cx}\frac {\pi^{\frac {n-1}2}}{\Gamma(\frac n2-1)}(4-x^2)^{\frac {n-1}2}dx$. Alpha will do the integral in terms of the incomplete Gamma function.