Green functions and conformal mapping
1 Introduction
The Green function for the Laplacian operator in two dimensions^{} is closely related to conformal mappings^{} to the unit disk. Given the Green function for a simplyconnected region with Dirichlet boundary conditions, one can construct the mapping by exponentiating the sum of the Green function and its conjugate harmonic function. In practise, this can be used to construct mapping functions^{} for various regions for which it is possible to solve the Dirichlet problem. In principle, it can be used to prove results about conformal mappings using the theory of differential equations. For instance, one can prove the Riemann mapping theorem^{} as a consequence of the existence of a solution to the Dirichlet problem.
2 Definition
Let $D$ be a simply connected subset of the complex plane with boundary $\beta \x88\x82\beta \x81\u2018D$ and let $a$ be a point in the interior of $D$. The Greenβs function is a function $g:D\beta \x86\x92\mathrm{\pi \x9d\x90\x91}$ such that

1.
$g=0$ on $\beta \x88\x82\beta \x81\u2018D$.

2.
${\beta \x88\x87}^{2}\beta \x81\u2018g=0$ on the interior of $D$.

3.
$g\beta \x81\u2019(z)\mathrm{log}\beta \x81\u2018za$ is bounded as $z$ approaches $a$.
Note: The third condition actually is equivalent^{} to the stronger condition that $g\beta \x81\u2019(z)\mathrm{log}\beta \x81\u2018za$ is analytic^{} at $a$. This follows from the general fact about harmonic functions.
3 Construction of the mapping function
Let $h$ be the conjugate harmonic function of to $g$. It can be shown that $h\beta \x81\u2019(z)\mathrm{arg}\beta \x81\u2018(za)$ is bounded as $z\beta \x86\x92a$ and, consequently, that $h$ is a multiplevalued function with branch point^{} at $a$ which increases by $2\beta \x81\u2019\mathrm{{\rm O}\x80}$ every time one encircles $a$.
Now consider the function $f$ defined as ${e}^{(g+i\beta \x81\u2019h)}$. This function is single valued because, when one circles about $a$, the argument of the exponential increases by $2\beta \x81\u2019\mathrm{{\rm O}\x80}\beta \x81\u2019i$, but adding $2\beta \x81\u2019\mathrm{{\rm O}\x80}\beta \x81\u2019i$ to an exponential does not change its value. Since $h$ is the conjugate harmonic function of $g$, it follows that $g+i\beta \x81\u2019h$ is holomorphic and, hence $f$ is also holomorphic.
Therefore, $f$ maps $D\beta \x88\x96\{a\}$ to $\mathrm{\pi \x9d\x90\x82}$. Various things can be said about this mapping.
Because of the maximum princliple, $g\beta \x81\u2019(z)>0$ for all $z$ in the interior of $D$. Hence, $f$ maps the interior of $D$ into the interior of the unit disk and maps $\beta \x88\x82\beta \x81\u2018D$ to the unit circle.
Furthermore, it can be shown that the function $f$ is invertible^{} so, in fact, it is a conformal diffeomorphism between $D$ and the unit disk.
Title  Green functions and conformal mapping 

Canonical name  GreenFunctionsAndConformalMapping 
Date of creation  20130322 15:57:06 
Last modified on  20130322 15:57:06 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  5 
Author  rspuzio (6075) 
Entry type  Topic 
Classification  msc 30A99 