In the part on Fourier analysis, we discuss pointwise convergence results, summability methods and, of course, convergence in the quadratic mean of Fourier series. More advanced topics include a first discussion of Hardy spaces. We also spend some time handling general orthogonal series expansions, in particular, related to orthogonal polynomials. Then we switch to the Fourier integral, i.e. the Fourier transform in Schwartz space, as well as in some Lebesgue spaces or of measures. Our treatment of ordinary differential equations starts with a discussion of some classical methods to obtain explicit integrals, followed by the existence theorems of Picard–Lindelö f and Peano which are proved by fixed point arguments. Linear systems are treated in great detail and we start a first discussion on boundary value problems. In particular, we look at Sturm–Liouville problems and orthogonal expansions. We also handle the hypergeometric differential equations (using complex methods) and their relations to special functions in mathematical physics. Some qualitative aspects are treated too, e.g. stability results (Ljapunov functions), phase diagrams, or flows. Our introduction to the calculus of variations includes a discussion of the Euler–Lagrange equations, the Legendre theory of necessary and sufficient conditions, and aspects of the Hamiliton–Jacobi theory. Related first order partial differential equations are treated in more detail. The text serves as a companion to lecture courses, also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions. Key Features: ○A coherent treatment of the entire core in mathematical analysis for undergraduate and graduate students — a unique source for learning as well as a reference text for later studies ○A comprehensive and modern treatment of two indispensible theories for every mathematician: Fourier Analysis. Ordinary Differential equations, Calculus of Variations ○More than 250 problems solved in full detail

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