Speaker
Description
Conformal mappings, i.e., complex differentiable bijective mappings, from one domain in the complex plane onto another domain preserve angles locally and therefore respect plane potential functions. This makes them useful in physics, for example in fluid mechanics and electrostatics. In particular, one is interested in finding a conformal mapping from a domain in the plane where a physical problem occurs to the complex unit circle since potential functions can easily be calculated here.
Bernhard Riemann (1826–1866) proved in [Riemann 1851, Art. 21] that such a mapping always exists if the starting domain is simply connected, i.e., has no holes. However, even if he took the plausibility for his proof from physics, he only showed the mere existence of the mapping without any concrete expression that could be used for explicit calculations.
As early as the winter semester 1863/64 Franz / Franciszek Mertens (1840–1927) remarked “that it was odd that Riemann had already proven the existence of a function which conformally maps, e.g., the area of a plane straight-sided triangle onto the area of a circle whereas for the present time the effective determination of such a function seems to surmount the powers of analysis” [Schwarz 1869, 105]. (Mertens was born in Schroda / Środa WielkoPoland. His father’s family came from Bremen. On his mother’s side, he had a Polish grandmother and a French grandfather. From 1865 to 1884 he was professor of mathematics at Kraków, where he lectured in Polish and published both in German and in Polish.) Hermann Amandus Schwarz (1843–1921) took up that problem and, encouraged by his academic teacher Karl Weierstraß (1815–1897), published an article with explicit formulas for conformal mappings in the case of polygons [Schwarz 1869]. In the meantime, also Elwin Bruno Christoffel (1829–1900) had attacked this problem [Christoffel 1867].
References
Christoffel, Elwin Bruno: Sul problema delle temperature stazionarie e la rappresentazione di una data superficie. Annali di Matematica Pura ed Applicata 1 (1867), 89–103.
Riemann, Bernhard: Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Ph.D. thesis. Göttingen 1851.
Schwarz, Hermann Amandus: Über einige Abbildungsaufgaben. Journal für die reine und angewandte Mathematik 70 (1869), 105–120.
