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Hardcover ISBN:  9781470452865 
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Book DetailsGraduate Studies in MathematicsVolume: 202; 2019; 480 ppMSC: Primary 30
In this text, the reader will learn that all the basic functions that arise in calculus—such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet—are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a selfcontained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here.
This book could serve for either a onesemester course or a twosemester course in complex analysis for beginning graduate students or for wellprepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus.
ReadershipGraduate students interested in complex analysis.

Table of Contents

Chapters

Basic calculus in the complex domain

Going deeper – the Cauchy integral theorem and consequences

Fourier analysis complex function theory

Residue calculus, the argument principle, and two very special functions

Conformal maps and geometrical aspects of complex function theory

Elliptic functions and elliptic integrals

Complex analysis and differential equations


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In this text, the reader will learn that all the basic functions that arise in calculus—such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet—are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a selfcontained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here.
This book could serve for either a onesemester course or a twosemester course in complex analysis for beginning graduate students or for wellprepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus.
Graduate students interested in complex analysis.

Chapters

Basic calculus in the complex domain

Going deeper – the Cauchy integral theorem and consequences

Fourier analysis complex function theory

Residue calculus, the argument principle, and two very special functions

Conformal maps and geometrical aspects of complex function theory

Elliptic functions and elliptic integrals

Complex analysis and differential equations