A Course in Mathematical Analysis: Volume 3, Complex Analysis, Measure and Integration: Volume 3
[Book Description]
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 the 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 instructors. Volume 1 focuses on the analysis of real-valued functions of a real variable. Volume 2 goes on to consider metric and topological spaces. This third volume develops the classical theory of functions of a complex variable. It carefully establishes the properties of the complex plane, including a proof of the Jordan curve theorem. Lebesgue measure is introduced, and is used as a model for other measure spaces, where the theory of integration is developed. The Radon-Nikodym theorem is proved, and the differentiation of measures discussed.
[Table of Contents]
Introduction ix
Part Five Complex analysis 625(176)
20 Holomorphic functions and analytic 627(23)
functions
20.1 Holomorphic functions 627(3)
20.2 The Cauchy--Riemann equations 630(5)
20.3 Analytic functions 635(6)
20.4 The exponential, logarithmic and 641(4)
circular functions
20.5 Infinite products 645(1)
20.6 The maximum modulus principle 646(4)
21 The topology of the complex plane 650(24)
21.1 Winding numbers 650(5)
21.2 Homotopic closed paths 655(6)
21.3 The Jordan curve theorem 661(6)
21.4 Surrounding a compact connected set 667(3)
21.5 Simply connected sets 670(4)
22 Complex integration 674(34)
22.1 Integration along a path 674(6)
22.2 Approximating path integrals 680(4)
22.3 Cauchy's theorem 684(5)
22.4 The Cauchy kernel 689(1)
22.5 The winding number as an integral 690(2)
22.6 Cauchy's integral formula for 692(6)
circular and square paths
22.7 Simply connected domains 698(1)
22.8 Liouville's theorem 699(1)
22.9 Cauchy's theorem revisited 700(2)
22.10 Cycles; Cauchy's integral formula 702(2)
revisited
22.11 Functions defined inside a contour 704(1)
22.12 The Schwarz reflection principle 705(3)
23 Zeros and singularities 708(25)
23.1 Zeros 708(2)
23.2 Laurent series 710(3)
23.3 Isolated singularities 713(5)
23.4 Meromorphic functions and the 718(2)
complex sphere
23.5 The residue theorem 720(4)
23.6 The principle of the argument 724(6)
23.7 Locating zeros 730(3)
24 The calculus of residues 733(16)
24.1 Calculating residues 733(1)
24.2 Integrals of the form ∫2π0 734(2)
f(cos t, sin t) dt
24.3 Integrals of the form 736(6)
∫∞--∞ f(x) dx
24.4 Integrals of the form ∫∞0 742(3)
xα f(x) dx
24.5 Integrals of the form ∫&inifin;0 745(4)
f(x) dx
25 Conformal transformations 749(19)
25.1 Introduction 749(1)
25.2 Univalent functions on C 750(1)
25.3 Univalent functions on the punctured 750(1)
plane C*
25.4 The Mobius group 751(7)
25.5 The conformal automorphisms of D 758(1)
25.6 Some more conformal transformations 759(4)
25.7 The space H (U) of holomorphic 763(2)
functions on a domain U
25.8 The Riemann mapping theorem 765(3)
26 Applications 768(33)
26.1 Jensen's formula 768(2)
26.2 The function π cot πz 770(2)
26.3 The functions πcosec πz 772(3)
26.4 Infinite products 775(3)
26.5 *Euler's product formula* 778(5)
26.6 Weierstrass products 783(7)
26.7 The gamma function revisited 790(4)
26.8 Bernoulli numbers, and the 794(3)
evaluation of ζ(2k)
26.9 The Riemann zeta function revisited 797(4)
Part Six Measure and Integration 801(135)
27 Lebesgue measure on R 803(14)
27.1 Introduction 803(1)
27.2 The size of open sets, and of closed 804(4)
sets
27.3 Inner and outer measure 808(2)
27.4 Lebesgue measurable sets 810(2)
27.5 Lebesgue measure on R 812(2)
27.6 A non-measurable set 814(3)
28 Measurable spaces and measurable 817(17)
functions
28.1 Some collections of sets 817(3)
28.2 Borel sets 820(2)
28.3 Measurable real-valued functions 822(3)
28.4 Measure spaces 825(4)
28.5 Null sets and Borel sets 829(1)
28.6 Almost sure convergence 830(4)
29 Integration 834(31)
29.1 Integrating non-negative functions 834(5)
29.2 Integrable functions 839(7)
29.3 Changing measures and changing 846(2)
variables
29.4 Convergence in measure 848(6)
29.5 The spaces L1R (X, Σ. μ) 854(2)
and L1C (X, Σ, μ)
29.6 The spaces LpR (X, Σ. μ) 856(7)
and LpC (X, Σ, μ), for 0 < p
< ∞
29.7 The spaces L∞R (X, Σ. 863(2)
μ) and L∞C (X, Σ, μ)
30 Constructing measures 865(19)
30.1 Outer measures 865(3)
30.2 Caratheodory's extension theorem 868(3)
30.3 Uniqueness 871(2)
30.4 Product measures 873(7)
30.5 Borel measures on R, I 880(4)
31 Signed measures and complex measures 884(12)
31.1 Signed measures 884(5)
31.2 Complex measures 889(2)
31.3 Functions of bounded variation 891(5)
32 Measures on metric spaces 896(7)
32.1 Borel measures on metric spaces 896(2)
32.2 Tight measures 898(2)
32.3 Radon measures 900(3)
33 Differentiation 903(12)
33.1 The Lebesgue decomposition theorem 903(3)
33.2 Sublinear mappings 906(2)
33.3 The Lebesgue differentiation theorem 908(4)
33.4 Borel measures on R, II 912(3)
34 Applications 915(21)
34.1 Bernstein polynomials 915(3)
34.2 The dual space of LpC (X, Σ, 918(1)
μ) for 1 & le; p < ∞
34.3 Convolution 919(5)
34.4 Fourier series revisited 924(3)
34.5 The Poisson kernel 927(7)
34.6 Boundary behaviour of harmonic 934(2)
functions
Index 936(4)
Contents For Volume I 940(3)
Contents For Volume II 943
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