Doctoral Dissertation

Abstract

At the end of November 1851 Riemann presented his doctoral dissertation, “Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Grösse” (Foundations of a general theory of functions of a complex variable). In order to appreciate the results obtained in the dissertation, it is useful to summarize briefly the results obtained up to that time, although it must be remem-bered that it is, unfortunately, very difficult to establish a clear notion of what Riemann himself really knew. A general feature of scholarly articles of that era is an extremely meager citation of sources, even in those cases when the borrowing is obvious and is not, strictly speaking, being hidden. For example, in Riemann’s work devoted to Abelian functions there is not a single reference to Niels Henrik Abel (1802–1829) and only one to Jacobi. In the case of Riemann’s dissertation his most significant predecessors in developing the theory of functions of a complex variable were Augustin-Louis Cauchy (1789–1857) and Karl Weierstrass (1815–1897). The fact that there were no references to the work of Weierstrass in the dissertation is not surprising: his main work had not yet been published, although it was known that it had been done. The situation with respect to Cauchy is different: by 1850 Cauchy had already published many papers on the theory of a complex variable.