Analysis III. Analytic and differential functions, manifolds and Riemann surfaces. Translated from the French by Urmie Ray (Q2261584)

Volume III of ``Analysis'' is a continuation of Volumes I and II, and consists of three chapters: \newline VIII - Cauchy Theory. \newline IX - Multivariate Differential and Integral Calculus. \newline X - The Riemann Surface of an Algebraic Function. A significant part of complex analysis has been already introduced in volume II, e.g., analyticity of the holomorphic functions, the maximum principle, functions analytic in an annulus, singular points, meromorphic functions, periodic holomorphic functions, the theorems of Liouville and of d'Alembert-Gauss, limits and infinite products of holomorphic functions, analytic functions defined by the Cauchy integral. In Chapter VIII, the author constructs integrals of holomorphic functions along curves in their domains of definition and then deduces the Cauchy integral formula. Applications of Cauchy's method, which are useful in the next chapters, include the Fourier transform, the Hankel integral, summation formulas, the Gamma function, the Mellin transform and others. The largest Chapter IX presents the differential calculus of multivariate functions, the concept of differential forms and their integration up to Stokes' formula for a \(p\)-dimensional path. The further paragraphs give a summary treatment of differential manifold theory with vector fields, differential operators, differential equations, and differential forms on manifolds. Integration of a differential form over an oriented manifold leads to the general Stokes' formula for differential forms of degree \(n-1\) on \(n\)-dimensional oriented manifolds \(X\), which holds for open sets \(\Omega\) with compact closure whose boundary \(\partial\Omega\) is an \((n-1)\)-dimensional submanifold of \(X\). The final Chapter X brings the readers to the notion of a Riemann surface and to the theory of holomorphic and meromorphic functions on Riemann surfaces. Algebraic functions and their Riemann surfaces are considered in detail. The textbook is of highest mathematical level. It is interesting that the author's style is rather non-typical for a book of this kind. Strict mathematical exposition is often neighboring upon pretty historical fragments sometimes of delicate contents. At the moment a passage looks like a fiction with long sentences but not an academic paper. Such surroundings make the mathematical statements of the book more romantic and unforgettable.