I am currently working on finding a Fourier series representation for the electric potential following an array of electrically charged, coaxial rings along their central axis. For $n\in\mathbb{N}$ and $R,d\in\mathbb{R}$, I was hoping to evaluate the following integral: $$a_n=\int\limits_{-\infty}^{+\infty}\frac{\cos\frac{n\pi z}{4d}}{\sqrt{R^2+z^2}}dz.$$ If it helps, $R=1$ and $d=0.75$. Is this integral solvable, or would a numerical analysis be of better use? (I need to show that $a_n$ drops sufficiently fast as to allow an approximation including only the 2nd and maybe the 4th or 6th coefficient. Odd-$n$ will drop out later anyway.)
Thank you for the help!!!!
Hint
$$I_n=\int\frac{\cos\left(\frac{n\pi }{4d}z\right)}{\sqrt{R^2+z^2}}dz$$ Let $z=x R$ to make $$I_n=\int\frac{\cos \left(\frac{\pi n R}{4 d}x\right)}{\sqrt{x^2+1}}dx=\int\frac{\cos(kx)}{\sqrt{x^2+1} } dx=\Re\Big[\int\frac{e^{ikx}}{\sqrt{x^2+1} } dx \Big]$$
Now, have a look here.