## Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Eine Abhandlung, zu deren öffentlicher Vertheidigung behuf Erlangung der Doctorwürde der Verfasser bereit sein wird am 16. December 1851

VERY RARE FIRST EDITION OF RIEMANN’S DISSERTATION: “ONE OF THE MOST IMPORTANT ACHIEVEMENTS OF 19TH CENTURY MATHEMATICS,” “WHICH MARKED A NEW ERA IN THE DEVELOPMENT OF THE THEORY OF ANALYTIC FUNCTIONS.”.

“Riemann’s doctoral thesis is, in short, a masterpiece,” introducing geometric and topological methods, notably the idea of a ‘Riemann surface’. It is also of great rarity, for “although [it] was a printed booklet, it was not usually published or publicised in the normal way; the candidate had to pay for the print-run, and sales and marketing were executed on an infinitesimal scale. So the first printing of Riemann’s thesis consisted only of the obligatory copies he had to hand in at Göttingen University, and a few copies for his personal use” (*Landmarks in Western Mathematics*, no. 34). The present copy is evidently one of those handed to the University. (Quotes above from Laugwitz; Kolmogorov & Yushkevich; and Derbyshire).

Riemann begins his thesis by offering a new foundation for the theory of analytic functions, based not on analytic expressions but on the assumption that the complex function *w = u + iv * of the complex variable *z = x + iy* is ‘differentiable’. Riemann noted that this condition is equivalent to requiring that *u* and *v* satisfy the ‘Cauchy-Riemann equations’ (as they are now called), and that, when the derivative is non-zero, it is also equivalent to requiring that the function determines a conformal mapping from the *z*-plane to the *w*-plane. At this point he refers to Gauss’s work on conformal mapping, published in 1825 in Schumacher’s *Astronomische Abhandlungen*, which he had studied in Berlin (this is the only reference in the thesis to the work of others).

In order to deal with multi-valued functions such as algebraic functions and their integrals, Riemann introduced the surfaces now named after him: the Riemann surface associated with a function is composed of as many sheets as there are branches of the function, connected in a particular way so that continuity is preserved and a single-valued function on the surface is obtained. Such a surface can be represented on a plane by a series of ‘cross-cuts’, which divide the surface into simply-connected regions. “Riemann’s thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces. It therefore introduced topological methods into complex function theory... Riemann’s thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces” (Mactutor).

The rest of the thesis is devoted to the study of functions on Riemann surfaces. From the Cauchy-Riemann equations it follows that if *w = u + iv* is an analytic function, then *u* and *v* are harmonic functions, i.e. solutions of Laplace’s equation. This establishes a link between the theory of analytic functions and potential theory, a subject with which Riemann was familiar, having attended Gauss and Weber’s Göttingen seminar on mathematical physics. (Gauss himself had made a decisive contribution to potential theory in 1849). Riemann’s approach in the remainder of the thesis was deeply influenced by potential theory.

To construct harmonic functions such as *u* and *v*, Riemann began with the case of a simply-connected region and made use of what he called ‘Dirichlet’s principle’ (he had learned it from Dirichlet’s lectures in Berlin): this asserts that the harmonic functions are exactly those which minimize the value of a certain integral. He then extended this to the non-simply connected case using cross-cuts and other variants. This approach was later to prove controversial, as Weierstrass gave examples of situations in which the minimizing function does not exist, but it was rehabilitated by Hilbert early in the next century.

The crowning glory of the thesis, and the most difficult part of the theory of conformal mappings, is his celebrated mapping theorem. “As an application of his approach he gave a ‘worked-out example’, showing that two simply-connected plane surfaces can always be made to correspond in such a way that each point of one corresponds continuously with its image in the other, and so that corresponding parts are ‘similar in the small’, or conformal ... what is nowadays called the ‘Riemann mapping theorem’.” (*Landmarks*, p. 454).

According to Richard Dedekind (*Bernhard Riemann’s Lebenslauf*, p. 7), Riemann probably conceived the main ideas of the thesis in autumn 1847. It was submitted on 14 November, 1851 and the Dean of the Faculty asked Gauss for his opinion. Always sparing with his praise, Gauss nevertheless wrote: “The paper submitted by Mr Riemann bears conclusive evidence of the profound and penetrating studies of the author in the area to which the topic dealt with belongs” (quoted from R. Remmert, “From Riemann surfaces to complex spaces”, *Bull. Soc. Math. France* (1998), p. 207). Following the thesis examination on 16 December, 1851, Riemann was awarded his *Doctor Philosophiae* and Gauss recommended that he be formally appointed to a position at Göttingen.

I. Grattan-Guiness, *Landmarks in Western Mathematics*, Chapter 34; Poggendorff II, 641; DSB XI 449-450; J. Derbyshire, *Prime Obsession*, 2003; A. N. Kolmogorov & A. P. Yushkevich (eds.), *Mathematics in the 19th century*, Vol. II, 1996; D. Laugwitz, *Bernhard Riemann, 1826-1866*, 1998.

Göttingen: E. A. Huth, 1851. Quarto (255 x 211 mm), pp [ii] 32. Stamps on title of the Göttingen Royal Observatory (of which Gauss was director from 1807 to 1855), and of Göttingen State and University Library (deaccessioned by librarian). The leaves contemporarily bound with green paper strip spine. Pencil-underlining to author's name, and another pencil annotation to upper right corner of front wrapper. Old library numbering in ink to upper left corner. EXCEEDINGLY RARE.

Price: $28,500 .