So the book I'm reading tells me to derive
\begin{align*} \mathcal{J}_c(m,n)&=\displaystyle\int_{-\infty+iy}^{\infty+iy}(cv)^{-k}e\left(\frac{-m}{c^2v}-nv\right)dv\\ &=\displaystyle\frac{2\pi}{i^kc}\left(\frac{n}{m}\right)^{\frac{k-1}{2}}\sum_{\ell=0}^{\infty}\frac{(-1)^\ell}{\ell!\Gamma(l+k)}\left(\frac{x}{2}\right)^{k+2\ell-1} \end{align*}
for a non-negative $m$ and $n>0$. The book provides $n=0$ as follows
$$\mathcal{J}_c(0,n)=\displaystyle\left(\frac{2\pi}{ic}\right)^k\frac{n^{k-1}}{\Gamma(k)}$$
and tells me to use this and power series expansion for $e(z)=e^{2\pi iz}$ to evaluate the above integral for $n>0$. I tried the calculation several times but couldn't get what I want. I know this can be very tedious but can someone help me out here?