A first course in complex analysis / Allan R. Willms.

Format:

Book

Author/Creator:

Willms, Allan R., author.

Series:

Synthesis Lectures on Mathematics and Statistics

Language:

English

Subjects (All):

Mathematics.

Physical Description:

1 online resource (237 pages)

Place of Publication:

Switzerland : Springer, [2022]

Summary:

This book introduces complex analysis and is appropriate for a first course in the subject at typically the third-year University level. It introduces the exponential function very early but does so rigorously. It covers the usual topics of functions, differentiation, analyticity, contour integration, the theorems of Cauchy and their many consequences, Taylor and Laurent series, residue theory, the computation of certain improper real integrals, and a brief introduction to conformal mapping. Throughout the text an emphasis is placed on geometric properties of complex numbers and visualization of complex mappings.

Contents:

Intro

Preface

Acknowledgments

Basics of Complex Numbers

Introduction

Cartesian and Polar Forms

Addition and Multiplication of Complex Numbers

Exercises

The Exponential Function

Euler's Formula

The Exponential as Polar Form

Conversion between Cartesian and Polar Forms

Conjugation

Integer and Rational Powers

Stereographic Projection

Functions of a Complex Variable

Set Terminology

Single-Valued and Multi-Valued Functions

Lines and Circles

Elementary Mappings of Lines and Circles

Visualizing Complex Functions

Some Elementary Functions

Polynomials

Rational Functions

Rational Powers

The Exponential

Trigonometric Functions

Hyperbolic Functions

The Logarithmic Function

Complex Powers

Inverse Trigonometric Functions

Inverse Hyperbolic Functions

Differentiation

The Derivative

Geometric Interpretation of the Derivative

The Cauchy-Riemann Equations

Sufficient Conditions for Differentiability

Other Forms of the Cauchy-Riemann Equations

Analytic Functions

Invertibility

Harmonic Functions

Singular Points

Isolated Singularities

Branch Points

Other Singularities

Riemann Surfaces

Contour Integration

Arcs, Contours, and Parameterizations

Definite Integrals and Derivatives of Parameterizations

An Application: Fourier Series

Contours

Contour Integrals

Cauchy Theory

The Cauchy-Goursat Theorem and its Consequences

Path Independence

Complex Extension of the Fundamental Theorem of Calculus

Path Deformation

The Cauchy Integral Formulas and their Consequences

Morera's Theorem

Cauchy's Inequality.

Liouville's Theorem

Fundamental Theorem of Algebra

Gauss' Mean Value Theorem

Maximum Modulus Theorem

Minimum Modulus Theorem

Poisson's Integral Formulas for the Circle and Half-Plane

Counting Zeros and Poles

Argument Theorem

Rouché's Theorem

Argument Principle

Series

Convergence

Sequences

Series Convergence Tests

Uniform Convergence Results

Power Series

Taylor Series

Zeros of Analytic Functions

Analytic Continuation

Laurent Series

Isolated Singularities Again

Residues

Calculation of Residues

The Residue Theorem

Calculation of Certain Real Integrals

Integrals of the Form _02 F(cos,sin) d

Improper Real Integrals

Conformal Mapping

Conformal Maps

Application to Laplace's Equation

Greek Alphabet

Answers to Selected Exercises

Author's Biography

Index.

Notes:

Includes index.

Description based on publisher supplied metadata and other sources.

Description based on print version record.

Other Format:

Print version: Willms, Allan R. A First Course in Complex Analysis

ISBN:

3-031-79176-2

OCLC:

1312163670