Doctoral Dissertation

Abstract

In 1851, at the end of November, Riemann presented his doctoral dissertation, “Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse” (Foundations of a general theory of functions of a complex variable). To appreciate the results obtained in the dissertation, it is useful to characterize briefly the results already obtained up to that time, although it should be pointed out that it is, unfortunately, very difficult to establish a clear notion of what Riemann himself really knew. What generally characterizes 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 dedicated to developing Abelian functions, there is not a single reference to Abel 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. The fact that there were no references to the work of Weierstrass in the dissertation is not surprising: he had not yet published his work, although it was known that it had been done. The situation with respect to Cauchy is different, however: by 1850 Cauchy had already published many works on the theory of a complex variable. His first significant work, “Mémoir sur la théorie des intégrales définies,” (Memoir on the theory of definite integrals) was communicated to the Paris Academy in 1814, but was published only in 1827 with several emendations reflecting the evolution of his own views.

“The shortest path between two truths in the real domain lies in the complex domain.”

J. Hadamard