CosmoSim is a simulator for gravitational lensing.
This is implemented in the SIE class (src/simlib/SIE.cpp).
To evaluate the lens potential $\psi^{\mathrm{R}}(x,y)$, we calculate the polar coordinates $R$ and $\phi$, and use the formula (see SIE) for $\psi_{\xi_0,f,\theta,D_\mathrm{L}}^\textrm{SIE(R)}(R,\phi)$.
This uses
\begin{aligned}
\frac{\partial\psi^{\mathrm{R}}}{\partial x} &=
C_0\frac{\sqrt{f}}{f’}\cdot
\left(
\cos\theta\cdot\sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x\cos\theta+y\sin\theta}{R}\right)
-\sin\theta\cdot\sin^{-1}\left(f’\cdot\frac{-x\sin\theta+y\cos\theta}{R}\right)
\right)
\\\
\frac{\partial\psi^{\mathrm{R}}}{\partial y} &=
C_0\frac{\sqrt{f}}{f’}\cdot
\left(
\sin\theta\cdot\sinh^{-1}\left(\frac{f’}{f}\cdot\frac{x\cos\theta+y\sin\theta}{R}\right)
+\cos\theta\cdot\sin^{-1}\left(f’\cdot\frac{-x\sin\theta+y\cos\theta}{R}\right)
\right)
\end{aligned}
as taken from the expression for $\vec\alpha(x,y)$ above.
Note in polar co-ordinates, we replace $x/R$ and $y/R$ with $\cos\phi$ and $\sin\phi$ respectively, and find that $\nabla\psi$ is constant in $R$.
Under Construction
\begin{align}
\alpha^m_s & =
\Gamma^m_s \square^{a^-}
\sum_{k=0}^{2k\le s} (-1)^k
\binom{s}{2k}
\frac{\partial^{s}}{(\partial x)^{s-2k}(\partial y)^{2k}}
\\\
\beta^m_s & =
\Gamma^m_s \square^{a^-}
\sum_{k=0}^{2k+1\le s} (-1)^k
\binom{s}{2k+1}
\frac{\partial^{s}}{(\partial x)^{s-2k-1}(\partial y)^{2k+1}}
\end{align}
where
\begin{equation}
\Gamma^m_s =
\begin{cases}
-(2^{-\delta_{0s}})\frac{\chi^{m+1}}{2^m}\binom{m+1}{(m+1-s)/2}, \quad\text{$m+s$ odd} \\\
0 \quad\text{$m+s$ even}
\end{cases}
\end{equation}
and
\begin{equation}
\square = (\nabla\psi)^2
\end{equation}
TODO $a^-$