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!!

You might combine the Legendre expansion (wlog $a>b$)
$$
\frac1{\sqrt{a^2+b^2-2ab\cos\varphi}}=\frac1a\sum_{n=0}^\infty\left(\frac ba\right)^nP_n(\cos\varphi)
$$
with the Fourier expansion (Gradshteyn-Ryzhik 8.826)
$$
P_n(\cos\varphi)=\\\frac{2^{n+2}n!}{\pi(2n+1)!!}\left(\sin(n+1)\varphi+\frac11\frac{n+1}{2n+3}\sin(n+3)\varphi+\frac11\frac32\frac{n+1}{2n+3}\frac{n+2}{2n+5}\sin(n+5)\varphi+\dots\right)
$$
and product-to-sum formulas when multiplying by $a-b\cos\varphi$.
