Green functions and conformal mapping
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 simply-connected 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.
on the interior of .
is bounded as approaches .
3 Construction of the mapping function
Let be the conjugate harmonic function of to . It can be shown that is bounded as and, consequently, that is a multiple-valued function with branch point at which increases by every time one encircles .
Now consider the function defined as . This function is single valued because, when one circles about , the argument of the exponential increases by , but adding to an exponential does not change its value. Since is the conjugate harmonic function of , it follows that is holomorphic and, hence is also holomorphic.
Therefore, maps to . Various things can be said about this mapping.
Because of the maximum princliple, for all in the interior of . Hence, maps the interior of into the interior of the unit disk and maps to the unit circle.
|Title||Green functions and conformal mapping|
|Date of creation||2013-03-22 15:57:06|
|Last modified on||2013-03-22 15:57:06|
|Last modified by||rspuzio (6075)|