How to evaluate $\int_{-1/2}^{1/2}e^{2\pi m i x}e^{\frac{2\pi n i}{x}}dx $?

Question

I'm interested in the integral

$$\int_{-1/2}^{1/2}e^{2\pi m i x}e^{\frac{2\pi n i}{x}}dx ,$$

where $m$ and $n$ are arbitrary nonnegative integers. Is there any possibility to evaluate this expression in a more or less explicit manner?

Answer 1

Let $x = \sqrt{\frac{n}{m}}\, z$. Then the integral becomes:

$$ I = \int_{-\frac{1}{2}\sqrt{\frac{m}{n}}}^{\frac{1}{2}\sqrt{\frac{m}{n}}}\exp\left(2\pi i\sqrt{mn}\left(x+\frac{1}{x}\right)\right)\,dx=2\int_{0}^{\frac{1}{2}\sqrt{\frac{m}{n}}}\cos\left(2\pi\sqrt{mn}\left(x+\frac{1}{x}\right)\right)\,dx$$ or: $$ I = 2\int_{0}^{2\arctan\left(\frac{1}{2}\sqrt{\frac{m}{n}}\right)}\cos\left(\frac{4\pi\sqrt{mn}}{\sin t}\right)\,\frac{dt}{1+\cos t}$$ or: $$ I = 2\int_{\frac{1}{2}\sqrt{\frac{m}{n}}+2\sqrt{\frac{n}{m}}}^{+\infty}\frac{\cos\left(4\pi\sqrt{mn}v\right)}{v^2-1+\sqrt{v^4-v^2}}\,dv$$ that is convergent by Dirichlet's criterion.