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# 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 $$z=x_1+ix_2$$, $$z_0=y_1+iy_2$$.

### Contributors

• Integrated by Justin Marshall.