I need solve this integral, and I tried various methods of solving and did not get it. The integral is:
$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$
where $t$ is a positive integer.
I need solve this integral, and I tried various methods of solving and did not get it. The integral is:
$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$
where $t$ is a positive integer.
This is the famous Poisson kernel (see http://en.wikipedia.org/wiki/Poisson_kernel). For example, ${\displaystyle \frac{1}{1 - 2t\cos\theta + t^2} = {1 \over 1 - t^2}Re\bigg(\frac{1 + te^{i\theta}}{1 - te^{i\theta}}\bigg)}$, so you're looking for the real part of ${1 \over 2\pi(1 - t^2)}\int_0^{2\pi}{\frac{1 + te^{i\theta}}{1 - te^{i\theta}}}d\theta$ As a complex integral this is ${1 \over 2\pi i(1 - t^2)}\int_{|z| = 1}\frac{1 + tz}{z(1 - tz)}\,dz$ By the Cauchy integral formula this evaluates to ${1 \over 1 - t^2}$ This is already real, so this is also the real part which is your answer.
HINT: Weierstrass Substitution
It is easier to use techniques from complex variables (residue theorem), substitute $ \cos(\theta) = \frac{z+\frac{1}{z}}{2} \,$ where $ z=\exp{(i\theta)}$ and integrate
$ \frac{1}{2\pi}\oint_{|z|=1}\frac{1}{1-2t\frac{z+\frac{1}{z}}{2} +t^2}d\theta = \dots $
$\frac{1}{1-2t\cos \theta +t^2}=\frac 1{1+t^2-2t\frac{(1-\tan^2\frac{\theta}2)}{(1+\tan^2\frac{\theta}2)}}$
$=\frac{\sec^2\frac{\theta}2}{(1+t^2)(1+\tan^2\frac{\theta}2)-2t(1-\tan^2\frac{\theta}2)}=\frac{\sec^2\frac{\theta}2}{(1-t)^2+\tan^2\frac{\theta}2(1+t)^2}$
$=\frac1{(1+t)^2}\frac{\sec^2\frac{\theta}2}{(\frac{1-t}{1+t})^2+\tan^2\frac{\theta}2}$
If $f(\theta)=\frac{1}{1-2t\cos \theta +t^2}, f(2\pi-\theta)=f(\theta)$,
So, $\int_{0}^{2\pi}f(\theta)d\theta=2\int_{0}^{\pi}f(\theta)d\theta$
$I=\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta =\frac{1}{\pi}\int_{0}^{\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$
$=\frac{1}{\pi(1+t)^2}\int_{0}^{\pi}\frac{\sec^2\frac{\theta}2}{(\frac{1-t}{1+t})^2+\tan^2\frac{\theta}2}d\theta$
Now we can put $z=\tan \frac{\theta}2$ in the given problem, if $\theta=0,z=0$ and if $\theta=\pi,z=∞$ and $dz=\frac{\sec^2\frac{\theta}{2}d\theta}{2}$
So, $I=\frac{1}{\pi(1+t)^2}\int_{0}^{∞}\frac{2dz}{(\frac{1-t}{1+t})^2+z^2}$
$=\frac{2}{\pi(1-t^2)} \tan^{-1}{\frac{(1+t)z}{1-t}} \mid_{0}^{∞}$
At $z=0, \tan^{-1}{\frac{(1+t)z}{1-t}}=0$ if $t \ne 1$
At $z=∞, \tan^{-1}{\frac{(1+t)z}{1-t}}=\frac{\pi}2$ if $ \frac{1+t}{1-t}>0 $ or if $ \frac{(1+t)(1-t)}{(1-t)^2}>0$ or if $1-t^2>0$ or if $-1< t< 1$
At $z=∞, \tan^{-1}{\frac{(1+t)z}{1-t}}=-\frac{\pi}2$ if $t>1$ or $t<-1$