Niels Jacob

Kristian P Evans

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-Lindelöf 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 Hamilton-Jacobi theory. Related first order partial differential equations are treated in more detail.The text serves as a companion to lecture courses, and it is also suitable for self-study. The text is complemented by ca. 260 problems with detailed solutions.

Course In Analysis, A - Vol. Iv: Fourier Analysis, Ordinary Differential Equations, Calculus Of Variations

Part 1 begins with an overview of properties of the real numbers and starts to introduce the notions of set theory. The absolute value and in particular inequalities are considered in great detail before functions and their basic properties are handled. From this the authors move to differential and integral calculus. Many examples are discussed. Proofs not depending on a deeper understanding of the completeness of the real numbers are provided. As a typical calculus module, this part is thought as an interface from school to university analysis. Part 2 returns to the structure of the real numbers, most of all to the problem of their completeness which is discussed in great depth. Once the completeness of the real line is settled the authors revisit the main results of Part 1 and provide complete proofs. Moreover they develop differential and integral calculus on a rigorous basis much further by discussing uniform convergence and the interchanging of limits, infinite series (including Taylor series) and infinite products, improper integrals and the gamma function. In addition they discussed in more detail as usual monotone and convex functions. Finally, the authors supply a number of Appendices, among them Appendices on basic mathematical logic, more on set theory, the Peano axioms and mathematical induction, and on further discussions of the completeness of the real numbers. Remarkably, Volume I contains ca. 360 problems with complete, detailed solutions.

Volume I: Introductory Calculus, Analysis of Functions of One Real Variable

A Course in Analysis

In this third volume of "A Course in Analysis", two topics indispensible for every mathematician are treated: Measure and Integration Theory; and Complex Function Theory. In the first part measurable spaces and measure spaces are introduced and Caratheodory's extension theorem is proved. This is followed by the construction of the integral with respect to a measure, in particular with respect to the Lebesgue measure in the Euclidean space. The Radon–Nikodym theorem and the transformation theorem are discussed and much care is taken to handle convergence theorems with applications, as well as Lp-spaces. Integration on product spaces and Fubini's theorem is a further topic as is the discussion of the relation between the Lebesgue integral and the Riemann integral. In addition to these standard topics we deal with the Hausdorff measure, convolutions of functions and measures including the Friedrichs mollifier, absolutely continuous functions and functions of bounded variation. The fundamental theorem of calculus is revisited, and we also look at Sard's theorem or the Riesz–Kolmogorov theorem on pre-compact sets in Lp-spaces. The text can serve as a companion to lectures, but it can also be used for self-studying. This volume includes more than 275 problems solved completely in detail which should help the student further. Contents: Measure and Integration Theory:First Look at σ-Fields and MeasuresExtending Pre-Measures. Carathéodory's TheoremThe Lebesgue-Borel Measure and Hausdorff MeasuresMeasurable MappingsIntegration with Respect to a Measure — The Lebesgue IntegralThe Radon-Nikodym Theorem and the Transformation TheoremAlmost Everywhere Statements, Convergence TheoremsApplications of the Convergence Theorems and MoreIntegration on Product Spaces and ApplicationsConvolutions of Functions and MeasuresDifferentiation RevisitedSelected TopicsComplex-Valued Functions of a Complex Variable:The Complex Numbers as a Complete FieldA Short Digression: Complex-Valued MappingsComplex Numbers and GeometryComplex-Valued Functions of a Complex VariableComplex DifferentiationSome Important FunctionsSome More TopologyLine Integrals of Complex-Valued FunctionsThe Cauchy Integral Theorem and Integral FormulaPower Series, Holomorphy and Differential EquationsFurther Properties of Holomorphic FunctionsMeromorphic FunctionsThe Residue TheoremThe Γ-Function, The ζ-Function and Dirichlet SeriesElliptic Integrals and Elliptic FunctionsThe Riemann Mapping TheoremPower Series in Several VariablesAppendices:More on Point Set TopologyMeasure Theory, Topology and Set TheoryMore on Möbius TransformationsBernoulli Numbers Readership: Undergraduate students in mathematics.

Kristian P Evans

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