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Mathematics LibreTexts

7.4.2: Green's Function and Conformal Mapping

For two-dimensional domains there is a beautiful connection between conformal mapping and Green's function. Let \(w=f(z)\) be a conformal mapping from a sufficiently regular connected domain in \(\mathbb{R}^2\) onto  the interior of the unit circle, see Figure 7.4.2.1


Figure 7.4.2.1: Conformal mapping

Then the Green function of \(\Omega\) is, see for example [16] or other text books about the theory of functions of one complex variable,
$$
G(z,z_0)=\frac{1}{2\pi}\ln\left|\frac{1-f(z)\overline{f(z_0)}}{f(z)-f(z_0)}\right|,
$$
where \(z=x_1+ix_2\), \(z_0=y_1+iy_2\).

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