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I am interested in computing the following integrals \begin{align} I (\mathbf{x}_{1}) {} & = \!\! \int_{\mathbf{D}} \!\! \mathrm{d} \mathbf{x} \, \frac{|\mathbf{x}_{1} \!-\! \mathbf{x}|}{\sqrt{1 \!-\! |\mathbf{x}|^{2}}} , \\ J (\mathbf{x}_{1} , \mathbf{x}_{2}) {} & = \!\! \int_{\mathbf{D}} \mathrm{d} \mathbf{x} \, \frac{|\mathbf{x}_{1} \!-\! \mathbf{x}| \, |\mathbf{x}_{2} \!-\! \mathbf{x}|}{\sqrt{1 \!-\! |\mathbf{x}|^{2}}} , \end{align} where the integrals are to be performed on the unit disc, ${ \mathbf{D} \!=\! \{ \mathbf{x} \!\in\! \mathbb{R}^{2} \, | \, |\mathbf{x}| \!\leq\! 1 \} }$, and the argument ${ \mathbf{x}_{1} , \mathbf{x}_{2} \!\in\! \mathbb{R^{2}} }$ can be both inside or outside the unit disc.

I have tried numerous approaches to perform this integral. For example, to compute ${ I(\mathbf{x}_{1}) }$ I tried:

(i) Writing a Fourier decomposition of the form \begin{align} |\mathbf{x}_{1} \!-\! \mathbf{x}| {} & = \mathrm{Max}[|\mathbf{x}_{1}|,|\mathbf{x}|] \, \sqrt{1 \!+\! \eta^{2} \!-\! 2 \eta \cos (\phi)} \nonumber \\ {} & = \sum_{\ell} \mathrm{e}^{\ell \phi} \, [\cdots] , \end{align} with $\phi$ the polar angle between $\mathbf{x}_{1}$ and $\mathbf{x}$, ${ 0 \!\leq\! \eta \!\leq\! 1 }$ given by ${ \eta \!=\! \mathrm{Min}[|\mathbf{x}_{1}|,|\mathbf{x}|]/\mathrm{Max}[|\mathbf{x}_{1}|,|\mathbf{x}|] }$, and the Fourier coefficients, ${ [\cdots] }$, involving elliptic integrals of the second kind.

(ii) Using the usual Legendre expansion of ${1/|\mathbf{x}_{1}\!-\!\mathbf{x}|}$, to write \begin{align} |\mathbf{x}_{1} \!-\! \mathbf{x}| {} & = \frac{|\mathbf{x}_{1} \!-\! \mathbf{x}|^{2}}{|\mathbf{x}_{1} \!-\! \mathbf{x}|} \nonumber \\ {} & = |\mathbf{x}_{1} \!-\! \mathbf{x}|^{2} \, \sum_{\ell} \eta^{\ell} \, P_{\ell} (\cos (\phi)) . \end{align} Unfortunately, these approaches did not prove much useful in evaluating explicitely the required integrals.

From another argument, I know that for ${ |\mathbf{x}_{1}| \!\leq\! 1 }$, one has \begin{equation} I (\mathbf{x}_{1}) = \tfrac{1}{2} \pi^{2} + \tfrac{1}{4} \pi^{2} |\mathbf{x}_{1}|^{2} , \end{equation} which I can reproduce by computing ${ I(\mathbf{x}_{1}) }$ numerically, but cannot recover analytically.

My question are therefore as follows:

  • How can one compute explicitly the integrals ${ I(\mathbf{x}_{1}) }$ and ${ J (\mathbf{x}_{1} , \mathbf{x}_{2}) }$ for arguments both inside and outside of the unit disc?
  • What is the appropriate "multipole expansion" that one should use to represent ${ |\mathbf{x}_{1} \!-\! \mathbf{x}| }$, so as to easily compute such integrals?
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