# Surface integral over a rectangle in Cartesian coordinates with singularity inside integration domain

I would like to integrate:

$f(\rho)=\int\limits_S{\frac{1}{R}dS}$ where $R=||\rho-\rho'||_2$ and $\rho'$ represents the distance from the bottom-left corner of the rectangle. i.e. if the integral is parametrized as:

$$\int\limits_0^Y\int\limits_0^X\frac{1}{R}dxdy,$$ then $\rho=x\hat{x}+y\hat{y}+0\hat{z}$ and $\rho=x_0\hat{x}+y_0\hat{y}+z_0\hat{z}$ represents an arbitrary point in 3D space which can be on the rectangle, on its boundary, near it, or far away, i.e.:

$$f(x_0,y_0,z_0)=\int\limits_0^Y\int\limits_0^X\frac{1}{\sqrt{(x_0-x)^2+(y_0-y)^2+z_0^2}}dxdy.$$

I am looking for an analytical/closed form solution to this integral. Thanks!

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Please refer to Hess and Smith "calculation of non-lifting potential flow about arbitrary three-dimensional bodies" , it's a classic problem in BEM. –  David Apr 26 '14 at 12:26