Jeremy Gray

The real and the complex : a history of analysis in the 19th century

Lagrange and foundations for the calculus.- Joseph Fourier.- Legendre.- Cauchy and continuity.- Cauchy: differentiation and integration.- Cauchy and complex functions to 1830.- Abel.- Jacobi.- Gauss.- Cauchy and complex function theory, 1830-1857.- Complex functions and elliptic integrals.- Revision.- Gauss, Green, and potential theory.- Dirichlet, potential theory, and Fourier series.- Riemann.- Riemann and complex function theory.- Riemanns later complex function theory.- Responses to Riemanns work.- Weierstrass.- Weierstrasss foundational results.- Revision { and assessment.- Uniform Convergence.- Integration and trigonometric series.- The fundamental theorem of the calculus.- The construction of the real numbers.- Implicit functions.- Towards Lebesgues theory of integration.- Cantor, set theory, and foundations.- Topology.- Assessment.

Weierstrass’s Foundational Results

Weierstrass is remembered for many specific discoveries in complex function theory, and here we consider two of them: his insight into the distinction between poles and essential singularities and the idea of a natural boundary of a complex function and the connection to nowhere differentiable real functions. We then look more briefly at his systematic presentation of the theory of elliptic functions, and content ourselves with a mention of his representation theorem, and his final account of a theory of Abelian functions.

Towards Lebesgue’s Theory of Integration

This chapter looks briefly at the Lebesgue integral, to see what questions it answered and what it is good for. The first half takes up two problems in analysis before 1900, the second half looks at how they, and the problem of the fundamental theorem of the calculus, were resolved using Lebesgue ’s theory.

Complex Functions and Elliptic Integrals

This chapter considers how elliptic functions and complex functions were first brought together. This was an important step for both subjects, which, as Jacobi noted in his lectures, seemed to be kept apart by the complications resulting from the two-valued nature of the integrand in the elliptic integrals.

Responses to Riemann’s Work

The impression is given in several places that few people responded quickly to Riemann ’s work, and those that did so with a view to finding fault with it, even arguing that it should be dismissed. It is true that some mathematicians began their papers by lamenting that nothing seems to have been done, but in fact his papers, with a few notable exceptions, drew a considerable amount of attention, as this chapter discusses.

Gauss, Green, and Potential Theory

The basic idea of a potential function is very simple. It applies to problems in dynamics where one is given a force, and the idea is to find a function f(x, y, z) whose partial derivatives give the components of a force. The function, if it exists, is called the potential function for the given force. It was introduced, without a name, by Lagrange in his study of gravitational attraction. The idea became even more importanst when Coulomb showed that electrostatic forces also obeyed an inverse-square law and then Ampere showed that there is also an inverse-square law in electrodynamics. It is in this fast-moving context that gauss and Green did their important theoretic work on potential functions in their 1830s and 1840s that is the subject of this chapter.

Integration and Trigonometric Series

Riemann’s previously unpublished Habilitation essay on the integration of trigonometric series was published for the first time in 1867, in an issue of the Gottingen Nachrichten that came out shortly after his death and carried several of his papers that he had left in almost a fit state to print. By then his reputation as a remarkable, but difficult, mathematician had spread, and what he wrote was read. This chapter considers some of the responses to his ideas about real functions and looks at the changes wrought in the relationship between continuity and differentiability, and between the implications of convergence and uniform convergence. This will lead us to look at the world of functions defined by series that do not converge uniformly, and were often considered in the late 19th century to be the most general kind of function, which is why they were often called ‘assumptionless functions’.

Riemann’s Later Complex Function Theory

Riemann continued to develop his own ideas, extending them to multiply-connected domains defined by algebraic curves. He presented them in public when he lectured on complex functions, in particular elliptic and Abelian functions , in 1855/56 and again in 1861/62, and he published them in his remarkable paper on abelian functions in 1857.

Cauchy and Continuity

One of the key figures in the transition from 18th century analysis to modern mathematical analysis is Augustin-Louis Cauchy. Over the next several chapters we shall look at various aspects of his contributions to real and complex analysis, and we shall see that his impact is very much tied up with the complicated, and political, nature of his life.

The Construction of the Real Numbers

Much of early 19th-century analysis rested uneasily on an intuitive notion of quantity that embraced all the measurable objects for which the natural numbers were inadequate, in particular lengths. Real analysis was the study of real varying quantities, and at least informally it was compatible with infinite and infinitesimal magnitudes. Cauchy had redefined these infinite and infinitesimal magnitudes in terms of limits, but it was still unclear what the domain of real quantities comprised, and it struck several German mathematicians that complications in real analysis might be alleviated by providing a good definition of what the real numbers might be. Here we look at Weierstrass’s not entirely successful account, then at Dedekind’s more elegant version, and finally and very briefly at the theories of Cantor and Heine.