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[SOLVED] Poisson kernel is even

dwsmith

Well-known member
Feb 1, 2012
1,673
For a fixed $r$ with $0\leq r < 1$, prove that $P(r,\theta)$ is an even function.


Take $-r$.
Then
\begin{alignat*}{3}
P(-r,\theta) & = & \frac{1}{2\pi}\frac{1 - (-r)^2}{1 - 2(-r)\cos\theta + (-r)^2}\\
& = & \frac{1}{2\pi}\frac{1 - r^2}{1 + 2r\cos\theta + r^2}
\end{alignat*}
I have $1 + 2r\cos\theta - r^2$. How can I get back the original denominator?
 

chisigma

Well-known member
Feb 13, 2012
1,704
For a fixed $r$ with $0\leq r < 1$, prove that $P(r,\theta)$ is an even function.
Because r is fixed the only variable remains $\theta$ and $cos \theta$ is an even function of $\theta$...

Kind regards

$\chi$ $\sigma$