I have a particular Linear Fractional Transformation from $\mathbb{C}$ to $\mathbb{C}$, I am using to solve Laplace's equation, and I was hoping to find the inverse of this transformation. Is there a algorithm that can be used to find the inverse of a Linear Fractional Transformation? Are there any useful algorithms for special cases?
2026-03-28 05:01:12.1774674072
Inverting a Linear Fractional Transformation
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The inverse of the linear fractional transformation $$z\to \frac{a z + b}{c z + d}$$ is the linear fractional transformation $$z\to \frac{e z + f}{g z + h}$$ where the matrix $$\begin{pmatrix}e & f\\ g & h\end{pmatrix}$$ is the inverse of the matrix $$\begin{pmatrix}a & b\\ c & d\end{pmatrix}.$$