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內(nèi)容概要

　　This book provides an introduction to complex analysis for students with some familiarity with complex numbers from high school. Students should be familiar with the Cartesian representation of complex numbers and with the algebra of complex numbers, that is, they should know that i2 = -1. A familiarity with multivariable calculus is also required, but here the fundamental ideas are reviewed. In fact, complex analysis provides a good training ground for multivariable calculus. It allows students to consolidate their understanding of parametrized curves, tangent vectors, arc length, gradients, line integrals, independence of path, and Green's theorem. The ideas surrounding independence of path are particularly difficult for students in calculus, and they are not absorbed by most students until they are seen again in other courses.

書籍目錄

PrefaceIntroductionFIRST PART Chapter 1 The Complex Plane and Elementary Functions    1.Complex Numbers    2.Polar Representation    3.Stereographic Projection    4.The Square and Square Root Functions    5.The Exponential Function　  6.The Logarithm Function　  7.Power Functions and Phase Factors　  8.Trigonometric and Hyperbolic Functions　Chapter 2 Analytic Functions　  1.Review of Basic Analysis　  2.Analytic Functions　  3.The CauChy-Riemann Equations　  4.Inverse Mappings and the Jacobian　  5.Harmonic Functions　  6.Conformal Mappings　  7.Fractional Linear Transformations　Chapter 3 Line Integrals and Harmonic Functions　  1.Line Integrals and Green's Theorem　  2.Independence of Path　  3.Harmonic Conjugates　  4.The Mean Value Property　  5.The Maximum Principle　  6.Applications to Fluid Dynamics　  7.Other Applications to Physics　Chapter 4 Complex Integration and Analyticity　  1.Complex Line Integrals　　2.Fundamental Theorem of Calculus for Analytic Functions　  3.Cauchy's Theorem  　4.The Cauchy Integral Formula　  5.Liouville's Theorem　  6.Morera's Theorem　  7.Goursat's Theorem　  8.Complex Notation and Pompeiu's Formula　Chapter 5 Power Series  　1.Infinite Series  　2.Sequences and Series of Functions　  3.Power Series　  4.Power Series Expansion of an Analytic Function　  5.Power Series Expansion at Infinity　  6.Manipulation of Power Series　  7.The Zeros of an Analytic Function　  8.Analytic Continuation　Chapter 6 Laurent Series and Isolated Singularities  　1.The Laurent Decomposition　  2.Isolated Singularities of an Analytic Function  　3.Isolated Singularity at Infinity    4.Partial Fractions Decomposition    5.Periodic Functions    6.Fourier Series　Chapter 7 The Residue Calculus    1.The Residue Theorem    2.Integrals Featuring Rational Functions    3.Integrals of Trigonometric Functions    4.Integrands with Branch Points    5.Fractional Residues    6.Principal Values    7.Jordan's Lemma    8.Exterior DomainsSECOND PART　Chapter 8 The Logarithmic Integral    1.The Argument Principle    2.Rouche's Theorem    3.Hurwitz's Theorem    4.Open Mapping and Inverse Function Theorems    5.Critical Points    6.Winding Numbers　……THIRD PARTHints and Solutions for Selected ExercisesReferencesList of SymbolsIndex

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