# Multipole-like integral on the unit disc

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 $$$$I (\mathbf{x}_{1}) = \tfrac{1}{2} \pi^{2} + \tfrac{1}{4} \pi^{2} |\mathbf{x}_{1}|^{2} ,$$$$ which I can reproduce by computing $${ 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?