History of the Riemann Mapping Theorem

Abstract

The Riemann mapping theorem, that an arbitrary simply connected region of the plane can be mapped one-to-one and conformally onto a circle, first appeared in the Inaugural dissertation of Riemann (1826-1866) in 1851. The theorem is important, for by it a result proved for the circle can often be transformed from the circle to a more general region. The proof is difficult, as involving both behavior of a function in the small (conformal mapping) and behavior in the large (one-toone mapping). Riemann's proof was open to criticism and in the following decades numerous mathematicians sought for a proof, e.g., H. A. Schwarz (1843-1921), A. Harnack (1851-1888), H. Poincare (1854-1912), etc., until the first rigorous proof was given in 1900 by W. F. Osgood. The proof of Osgood represented, in my opinion, the coming of age of mathematics in America. Until then, numerous American mathematicians had gone to Europe for their doctorates, or for other advanced study, as indeed did Osgood. But the mathematical productivity in this country in quality lagged behind that of Europe, and no American before 1900 had reached the heights that Osgood then reached. William Fogg Osgood (1864-1943) was born in Boston in 1864, graduated from Harvard College in 1886, stayed in Cambridge for a year of graduate work, and then went to Gottingen with a Harvard fellowship for further study, especially with Felix Klein (1849-1925). According to gossip, Osgood became so enamored of a Gottingen lady that his work suffered and Klein sent him to Erlangen for his doctorate. In any case, he was accorded the degree from Erlangen in 1890 for a thesis on Abelian integrals, and one or two days later he married the girl in G6ttingen, and one or two days still later they sailed for the United States of America. His