Integration of exponential and square root function
I need to solve this $$\int_{-\pi}^{\pi} \frac{e^{ixn}}{\sqrt{x^2+a^2}}\,dx,$$ where $i^2=-1$ and $a$ is a constant.
Solution 1:
This is not complete yet.
$$\int_{-\pi}^{\pi} \dfrac {\exp(inx)}{\sqrt{x^2+a^2}}dx=\int_{-\pi}^{\pi}\dfrac {\cos(nx)}{\sqrt{x^2+a^2}}dx+i \int_{-\pi}^{\pi}\dfrac {\sin(nx)}{\sqrt{x^2+a^2}}dx=2\int_{0}^{\pi}\dfrac {\cos(nx)}{\sqrt{x^2+a^2}}dx=2I$$
Set $x=a \sinh y$ then $dx=(a \cosh y) dy$ and $\sqrt{x^2+a^2}=a \cosh y$. Let $\pi=a \sinh y_0$. Then the integration $I$ becomes: $$I =\int_{0}^{y_0}\cos(n a \sinh y)dy$$