## Conformal mappings and the Schwarz-Christoffel transformation

##### Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Let ? be an open and connected set in the complex plane. A mapping f : ? ? C is said to be conformal at a point z0 if it preserves angles and orientation between curves intersecting at z0. We discuss tangent lines and their relationship to the conformal property. This property is useful for determining the points at which a function is conformal. We provide several examples and a summary of boundary behavior properties. We discuss two types of conformal mappings, linear fractional transformations and the Schwarz-Christoffel transformation. Linear fractional transformations map lines and circles to either lines or circles; meanwhile, the Schwarz-Christoffel transformation maps the upper half-plane onto a polygon P. We prove several properties of these mappings and demonstrate their importance with the use of examples. These results are useful for solving boundary value problems on irregular regions. If we map such a region onto another region whose solution is known, we are able to then obtain a solution on the first region. Important physical applications of this thesis include determination of isotherms, streamlines, and air gap reluctance for magnetic circuits

##### Degree

M.A.