I wish to solve the following definite integral:
$\int^{b}_a P_r(\theta-\phi) d\phi$
,where $P_r(\theta-\phi) = \frac{1-r^2}{1 + r^2 -2r cos(\theta - \phi)}$ is the Poisson kernel. This is something I have come across while solving the Laplace equation on a unit disk in the context of electrostatics. The definite integral of this equation can be calculated and the solution obtained using Mathematica (and other online integral calculators) is given by:
$2 tan^{-1}\Big((\frac{1+r}{1-r})tan(\frac{\theta - \phi}{2})\Big) + constant$
. I believe that the definite integral is a bit tricky since the range of the inverse tangent function can be $[-\pi/2, \pi/2]$, $[\pi/2, 3\pi/2]$,... and so on. When trying to find the definite integral of the same function using Mathematica or other tools, I keep getting the message that the computation time has exceeded. I assumed for a while that the integral might just be:
$2 tan^{-1}\Big((\frac{1+r}{1-r})tan(\frac{\theta - b}{2})\Big) - 2 tan^{-1}\Big((\frac{1+r}{1-r})tan(\frac{\theta - a}{2})\Big)$
, but I am not sure if it is the answer. I have used other online tools as well but all of them are giving the same error for some reason.
