# $\frac1{2\pi\rho^n}\int_{-\pi}^{\pi}\exp\left(\frac{2+\rho\cos\theta}{4+4\rho\cos\theta+\rho^2}\right)\cos\beta_nd\theta$ does not depend on $\rho$.

Let

$$\beta_n=\frac{\rho\sin\theta}{4+4\rho\cos\theta+\rho^2}+n\theta$$

where $0<\rho<2$. Could anyone give me some hints to prove analytically that

$$\frac1{2\pi\rho^n}\int_{-\pi}^{\pi}\exp\left(\frac{2+\rho\cos\theta}{4+4\rho\cos\theta+\rho^2}\right)\cos\beta_nd\theta$$

does not depend on $\rho$?

• First instinct is take derivative with respect to $\rho$ – Simply Beautiful Art Jun 3 '17 at 15:44

Therefore the integral does depend on $\rho$. (Well, it does not depend on $\rho$ if $n = 0$.)