How to prove $(\frac1\pi)\int_{-\pi}^\pi \left(\frac {1-r^2}{1-2r\cos\theta+r^2}\right)^2d\theta = \frac {2(1+r^2)}{1-r^2}$?

Question

So for this problem I need to use the fact that $\frac {1-r^2}{1-2r\cos\theta+r^2}$=$1+2\sum_{n=1}^{\infty} r^n\cos n\theta$.

I replaced the term in the integral but i ended up getting $\sum_{n=1}^{\infty} r^n+2$ as my final answer. Im not sure if I got it right and i need to reframe it to get $\frac {2(1+r^2)}{1-r^2}$ or my answer is completely wrong.

Thanks for the help.

Answer 1

A much simpler way is to note that $$\int_{0}^{\pi}\frac{dx}{a+b\cos x} =\frac{\pi} {\sqrt{a^2-b^2}}, a>|b|$$ and differentiate it with respect to $a$ to get $$\int_{0}^{\pi}\frac{dx}{(a+b\cos x) ^2}=\frac{\pi a} {(a^2-b^2)^{3/2}}$$ Putting $a=1+r^2,b=-2r$ and noting that $$a^2-b^2=(1-r^2)^2$$ we can see that $$\int_{0}^{\pi}\frac{dx}{(1-2r\cos x+r^2)^2}=\frac{\pi(1+r^2)}{(1-r^2)^3}$$ which is equivalent to your result in question.

The integral at the beginning of this answer is evaluated by using non-obvious substitution $$(a+b\cos x) (a-b\cos t) =a^2-b^2$$

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Your own approach should lead to $$\int_{0}^{\pi}\left(\frac{1-r^2}{1-2r\cos x+r^2}\right)^2\,dx=\pi+4\sum_{n=1}^{\infty}r^{2n}\int_{0}^{\pi}\cos^2 nx\, dx$$ which equals $$\pi+2\pi\sum_{n=1}^{\infty}r^{2n}=\frac{\pi(1+r^2)}{1-r^2}$$ Note that while squaring you also get terms like $\cos mx\cos nx$ but their integral vanishes so I have ommitted these in the above calculation.

Also $x$ has been used in place of $\theta$ to reduce typing effort.

Answer 2

You have reduced the problem to evaluating $$ \frac{1}{\pi}\int_{-\pi}^{\pi}\left(1+2\sum_{n=1}^{\infty}r^n\cos(n\theta)\right)^2d\theta. $$ The terms $1,\cos(\theta),\cos(2\theta),\cdots$ are mutually orthogonal on $[-\pi,\pi]$, which further reduces the above to $$ \frac{1}{\pi}\left(\int_{-\pi}^{\pi}1d\theta+4\sum_{n=1}^{\infty}r^{2n}\int_{-\pi}^{\pi}\cos^2(n\theta)d\theta\right) \\ = 2+\frac{4}{\pi}\sum_{n=1}^{\infty}r^{2n}\pi \\ =2+4r^{2}\sum_{n=0}^{\infty}(r^2)^n \\ =2+4r^{2}\frac{1}{1-r^2} \\ =2\frac{1+r^2}{1-r^2}. $$ And that checks with your answer.