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Fix $r\in(0,2)$, and consider arbitrary points $\mathbf{x},\mathbf{q}\in\mathbf{S}^2\subseteq\mathbf{R}^{3}$ such that $q_z > 1-r$ and $x_z > 1-r$. Consider the path $\gamma:[0,2\pi)\to\mathbf{S}^2$ given by $$\gamma(t)=(R\cos(t),R\sin(t),1-r),$$ where $R=\sqrt{r(2-r)}$. I want to compute $$\int_{\gamma([0,2\pi])}\frac{\ln(1-\mathbf{y}\cdot\mathbf{q})}{1-\mathbf{x}\cdot\mathbf{y}}~d\ell(\mathbf{y}).$$ I've been having a lot of trouble actually working this out by hand, and I'm not sure if there are any clever techniques that I could use to evaluate the above expression. I came across this integral when working out a representation formula for a solution to some sort of Dirichlet problem, and it would be very nice to have a closed form expression of the above.

If there isn't any nice closed form solution for such an expression, any ideas on numerical techniques for computing the above integral quickly would be also appreciated.

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  • $\begingroup$ $x$ and $q$ are essentially two parameters. Have you tried to use the definition of line integral? $\endgroup$ – Matheman Jun 21 '17 at 21:41
  • $\begingroup$ I understand that $x$ and $q$ are just two parameters. I also did use the definition of the line integral to expand it out as a definite integral from $0$ to $2\pi$, and that's where I've actually been having trouble evaluating the integral. $\endgroup$ – yousuf soliman Jun 22 '17 at 12:19
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First, consider your sphere as a subset of the Riemann sphere, then take the inverse transformation to take your curve from the sphere to the complex plane, then compute the line integral as usual.

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