Help to find complex Fourier series coefficient of this periodic function I'm having big trouble finding the complex Fourier series coefficient of the following periodic function
$$\frac{a-b\cos\varphi}{\sqrt{a^2+b^2-2ab\cos\varphi}}$$
Mathematica is unable to compute it!!

$$\frac{1}{\pi} \int_0^{\pi} \frac{a-b\cos\varphi}{\sqrt{a^2+b^2-2ab\cos\varphi}} \cos n \varphi\, d\varphi =  \, _3\tilde{F}_2\left(\tfrac{1}{2},\tfrac{1}{2},1;1-n,n+1;\tfrac{4 a b}{(a+b)^2}\right)$$
$$\qquad\qquad\qquad\qquad-\frac{ b}{a+b} \, _3\tilde{F}_2\left(\tfrac{1}{2},\tfrac{3}{2},2;2-n,n+2;\tfrac{4 a b}{(a+b)^2}\right),$$
with $\tilde{F}$ defined here.
