Conformal mapping

Physics and engineering

Conformal mappings are invaluable for solving problems in engineering and physics that can be expressed in terms of functions of a complex variable yet exhibit inconvenient geometries. By choosing an appropriate mapping, the analyst can transform the inconvenient geometry into a much more convenient one. For example, one may wish to calculate the electric field, ${\displaystyle E(z)}$, arising from a point charge located near the corner of two conducting planes separated by a certain angle (where ${\displaystyle z}$ is the complex coordinate of a point in 2-space). This problem per se is quite clumsy to solve in closed form. However, by employing a very simple conformal mapping, the inconvenient angle is mapped to one of precisely ${\displaystyle \pi }$ radians, meaning that the corner of two planes is transformed to a straight line. In this new domain, the problem (that of calculating the electric field impressed by a point charge located near a conducting wall) is quite easy to solve. The solution is obtained in this domain, ${\displaystyle E(w)}$, and then mapped back to the original domain by noting that ${\displaystyle w}$ was obtained as a function (viz., the composition of ${\displaystyle E}$ and ${\displaystyle w}$) of ${\displaystyle z}$, whence ${\displaystyle E(w)}$ can be viewed as ${\displaystyle E(w(z))}$, which is a function of ${\displaystyle z}$, the original coordinate basis. Note that this application is not a contradiction to the fact that conformal mappings preserve angles, they do so only for points in the interior of their domain, and not at the boundary. Another example is the application of conformal mapping technique for solving the boundary value problem of liquid sloshing in tanks.