The three volumes of A Course in Mathematical Analysis provide a full and detailed account of all those elements of real and complex analysis that an undergraduate mathematics student can expect to encounter in their first two or three years of study. Containing hundreds of exercises, examples and applications, these books will become an invaluable resource for both students and teachers. Volume I focuses on the analysis of realvalued functions of a real variable. Volume II goes on to consider metric and topological spaces. This third volume covers complex analysis and the theory of measure and integration.

Introduction page ix

Part Five Complex analysis 625

20 Holomorphic functions and analytic functions 627

      20.1 Holomorphic functions 627

      20.2 The Cauchy-Riemann equations 630

      20.3 Analytic functions 635

      20.4 The exponential,logarithmic and circular functions 641

      20.5 Infinite products 645

      20.6 The maximum modulus principle 646

21 The topology of the complex plane 650

      21.1 Winding numbers 650

      21.2 Homotopic closed paths 655

      21.3 The Jordan curve theorem 661

      21.4 Surrounding a compact connected set 667

      21.5 Simply connected sets 670

22 Complex integration 674

      22.1 Integration along a path 674

      22.2 Approximating path integrals 680

      22.3 Cauchy's theorem 684

      22.4 The Cauchy kernel 689

      22.5 The winding number as an integral 690

      22.6 Cauchy's integral formula for circular and square paths 692

      22.7 Simply connected domains 698

      22.8 Liouville's theorem 699

      22.9 Cauchy's theorem revisited 700

      22.10 Cycles; Cauchy's integral formula revisited 702

      22.11 Functions defined inside a contour 704

      22.12 The Schwarz reflection principle 705

23 Zeros and singularities 708

      23.1 Zeros 708

      23.2 Laurent series 710

      23.3 Isolated singularities 713

      23.4 Meromorphic functions and the complex sphere 718

      23.5 The residue theorem 720

      23.6 The principle of the argument 724

      23.7 Locating zeros 730

24 The calculus of residues 733

      24.1 Calculating residues 733

      24.2 Integrals of the form ∫2π 0 ƒ(cos t, sin t) dt 734

      24.3 Integrals of the form ∫∞ -∞ ƒ(x) dx 736

      24.4 Integrals of the form ∫∞ 0 xɑ ƒ(x) dx 742

      24.5 Integrals of the form ∫∞ 0 ƒ(x) dx 745

25 Conformal transformations 749

      25.1 Introduction 749

      25.2 Univalent functions on C 750

      25.3 Univalent functions on the punctured plane C* 750

      25.4 The Möbius group 751

      25.5 The conformal automorphisms of D 758

      25.6 Some more conformal transformations 759

      25.7 The space H(U) of holomorphic functions on a domain U 763

      25.8 The Riemann mapping theorem 765

26 Applications 768

26.1 Jensen's formula 768

26.2 The function π cot πz 770

26.3 The functions π cosec πz 772

26.4 Infinite products 775

26.5 *Euler's product formula* 778

26.6 Weierstrass products 783

26.7 The gamma function revisited 790

26.8 Bernoulli numbers, and the evaluation of ζ(2k) 794

26.9 The Riemann zeta function revisited 797

Part Six Measure and Integration 801

27 Lebesgue measure on R 803

      27.1 Introduction 803

      27.2 The size of open sets, and of closed sets 804

      27.3 Inner and outer measure 808

      27.4 Lebesgue measurable sets 810

      27.5 Lebesgue measure on R 812

      27.6 A non-measurable set 814

28 Measurable spaces and measurable functions 817

      28.1 Some collections of sets 817

      28.2 Borel sets 820

      28.3 Measurable real-valued functions 822

      28.4 Measure spaces 825

      28.5 Null sets and Borel sets 829

      28.6 Almost sure convergence 830

29 Integration 834

      29.1 Integrating non-negative functions 834

      29.2 Integrable functions 839

      29.3 Changing measures and changing variables 846

      29.4 Convergence in measure 848

      29.5 The spaces L1 R (X, ∑, μ) and L1 C (X, ∑, μ) 854

      29.6 The spaces Lp R (X, ∑, μ) and Lp C (X, ∑, μ) for 0

      29.7 The spaces L∞ R (X, ∑, μ) and L∞ C (X, ∑, μ) 863

30 Constructing measures 865

      30.1 Outer measures 865

      30.2 Caratheodory's extension theorem 868

      30.3 Uniqueness 871

      30.4 Product measures 873

      30.5 Borel measures on R, I 880

31 Signed measures and complex measures 884

      31.1 Signed measures 884

      31.2 Complex measures 889

      31.3 Functions of bounded variation 891

32 Measures on metric spaces 896

      32.1 Borel measures on metric spaces 896

      32.2 Tight measures 898

      32.3 Radon measures 900

33 Differentiation 903

      33.1 The Lebesgue decomposition theorem 903

      33.2 Sublinear mappings 906

      33.3 The Lebesgue differentiation theorem 908

      33.4 Borel measures on R, II 912

34 Applications 915

      34.1 Bernstein polynomials 915

      34.2 The dual space of Lp C (X, ∑, μ), for 0≤p<∞ 918

      34.3 Convolution 919

      34.4 Fourier series revisited 924

      34.5 The Poisson kernel 927

      34.6 Boundary behaviour of harmonic functions 934

Index 936

Contents for Volume I 940

Contents for Volume II 943

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