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Integrals

The following integrals are required for calculating the necessary Green's functions of a single voxel. They are contained in [13]. Given $ r = \sqrt{x^2 + y^2 + z^2}$

$\displaystyle \int \frac{1}{r} \, dy$ $\displaystyle =$ $\displaystyle \mathrm{sinh}^{-1}\left(\frac{y}{\sqrt{x^2 + z^2}}\right)$ (29)
$\displaystyle \iint \frac{1}{r^3} \, dx \, dy$ $\displaystyle =$ $\displaystyle \frac{1}{z} \mathrm{atan}\left(\frac{xy}{zr}\right)$ (30)
$\displaystyle \iint \frac{1}{r} \, dx \, dy$ $\displaystyle =$ $\displaystyle y \, \mathrm{sinh}^{-1}\left(\frac{x}{\sqrt{y^2 + z^2}}\right) +x... ...frac{y}{\sqrt{x^2 + z^2}}\right) - z \, \mathrm{atan}\left(\frac{xy}{zr}\right)$ (31)