Undergraduate Analysis is a logically self-contained introduction to analysis, designed for students who have completed two years of calculus. This text focuses on key aspects of uniform convergence and uniform limits within the realms of differentiation and integration. The book covers a range of topics including the classical test for convergence of series, Fourier series, polynomial approximation, and the Poisson kernel. It further explores the construction of harmonic functions on the disc, ordinary differential equations, curve integrals, derivatives in vector spaces, and multiple integrals. In this second edition, Serge Lang expands the original framework with new material and revisions throughout. A fresh chapter on locally integrable vector fields is introduced, along with enhanced sections on heat kernels in connection with Dirac families and the completion of normed vector spaces. Numerous exercises enrich the exploration of these fundamental analytical concepts.
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