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El. knyga: Introduction to Complex Analysis and the Laplace Transform [Taylor & Francis e-book]

(Indiana University, IN, USA.)
  • Formatas: 384 pages, 112 Line drawings, black and white; 112 Illustrations, black and white
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 21-Dec-2021
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9780367810283
Kitos knygos pagal šią temą:
  • Taylor & Francis e-book
  • Kaina: 170,80 €*
  • * this price gives unlimited concurrent access for unlimited time
  • Standartinė kaina: 244,00 €
  • Sutaupote 30%
Introduction to Complex Analysis and the Laplace TransformDidesnis paveikslėlis
  • Formatas: 384 pages, 112 Line drawings, black and white; 112 Illustrations, black and white
  • Serija: Textbooks in Mathematics
  • Išleidimo metai: 21-Dec-2021
  • Leidėjas: Chapman & Hall/CRC
  • ISBN-13: 9780367810283
Kitos knygos pagal šią temą:
Taylor & Francis e-booksMore info about Taylor & Francis e-books
Partnerių interneto svetainė: https://www.taylorfrancis.com/books/9780367810283
The aim of this comparatively short textbook is a sufficiently full exposition of the fundamentals of the theory of functions of a complex variable to prepare the student for various applications. Several important applications in physics and engineering are considered in the book.

This thorough presentation includes all theorems (with a few exceptions) presented with proofs. No previous exposure to complex numbers is assumed. The textbook can be used in one-semester or two-semester courses.

In one respect this book is larger than usual, namely in the number of detailed solutions of typical problems. This, together with various problems, makes the book useful both for self- study and for the instructor as well.

A specific point of the book is the inclusion of the Laplace transform. These two topics are closely related. Concepts in complex analysis are needed to formulate and prove basic theorems in Laplace transforms, such as the inverse Laplace transform formula. Methods of complex analysis provide solutions for problems involving Laplace transforms.

Complex numbers lend clarity and completion to some areas of classical analysis. These numbers found important applications not only in the mathematical theory, but in the mathematical descriptions of processes in physics and engineering.
Preface ix
Author xi
Introduction xiii
1 Complex Numbers and Their Arithmetic
1(18)
1.1 Complex Numbers
1(2)
1.2 Operations with Complex Numbers
3(16)
2 Functions of a Complex Variable
19(14)
2.1 The Complex Plane
19(5)
2.1.1 Curves in the complex plane
19(1)
2.1.2 Domains
20(4)
2.2 Sequences of Complex Numbers and Their Limits
24(3)
2.3 Functions of a Complex Variable; Limits and Continuity
27(6)
3 Differentiation of Functions of a Complex Variable
33(14)
3.1 The Derivative. Cauchy-Riemann Conditions
33(7)
3.1.1 The derivative and the differential
33(2)
3.1.2 Cauchy-Riemann conditions
35(3)
3.1.3 Analytic functions
38(2)
3.2 The Connection between Analytic and Harmonic Functions
40(3)
3.3 The Geometric Meaning of the Derivative. Conformal Mappings
43(4)
3.3.1 The geometric meaning of the argument of the derivative
43(2)
3.3.2 The geometric meaning of the modulus of the derivative
45(1)
3.3.3 Conformal mappings
46(1)
4 Conformal Mappings
47(38)
4.1 Linear and Mobius Transformations
47(13)
4.1.1 Linear functions
47(3)
4.1.2 Mobius transformations
50(10)
4.2 The Power Function. The Concept of Riemann Surface
60(6)
4.3 Exponential and Logarithmic Functions
66(6)
4.3.1 Exponential function
66(2)
4.3.2 The logarithmic function
68(4)
4.4 Power, Trigonometric, and Other Functions
72(8)
4.4.1 The general power function
72(2)
4.4.2 The trigonometric functions
74(2)
4.4.3 Inverse trig functions
76(1)
4.4.4 The Zhukovsky function
77(3)
4.5 General Properties of Conformal Mappings
80(5)
5 Integration
85(30)
5.1 Definition of the Contour Integral
85(6)
5.1.1 Properties of the contour integral
88(3)
5.2 Cauchy-Goursat Theorem
91(5)
5.3 Indefinite Integral
96(5)
5.4 The Cauchy Integral Formula
101(14)
6 Series
115(52)
6.1 Definitions
115(5)
6.2 Function Series
120(7)
6.3 Power Series
127(7)
6.4 Power Series Expansion
134(9)
6.5 Uniqueness Property
143(5)
6.6 Analytic Continuations
148(9)
6.7 Laurent Series
157(10)
7 Residue Theory
167(44)
7.1 Isolated Singularities
167(13)
7.2 Residues
180(9)
7.3 Computing Integrals with Residues
189(12)
7.3.1 Integrals over closed curves
190(1)
7.3.2 Real integrals of the form ∞2π0(cos φ, sin φ) dφ, where R is a rational function of cos φ and sin φ
191(2)
7.3.3 Improper integrals
193(8)
7.4 Logarithmic Residues and the Argument Principle
201(10)
8 Applications
211(54)
8.1 The Schwarz-Christoffel Transformation
211(15)
8.2 Hydrodynamics. Simply-connected Domains
226(11)
8.2.1 Complex potential of a vector field
227(2)
8.2.2 Simply-connected domains
229(8)
8.3 Sources and Sinks. Flow around Obstacles
237(19)
8.3.1 Sources and sinks
237(1)
8.3.2 Vortices
238(6)
8.3.3 Flow around obstacles
244(6)
8.3.4 The Zhukovsky airfoils
250(3)
8.3.5 Lifting force
253(3)
8.4 Other Interpretations of Vector Fields
256(9)
8.4.1 Electrostatics
256(5)
8.4.2 Heat flow
261(1)
8.4.3 Remarks on boundary value problems
262(3)
9 The Laplace Transform
265(42)
9.1 The Laplace Transform
266(8)
9.2 Properties of the Laplace Transformation
274(15)
9.3 Applications to Differential Equations
289(18)
9.3.1 Linear ODEs
289(2)
9.3.2 Finding the original function from its transform
291(5)
9.3.3 Differential equations with piecewise defined right hand sides
296(3)
9.3.4 Application of the convolution operation to solving differential equations
299(3)
9.3.5 Systems of differential equations
302(5)
Solutions, Hints, and Answers to Selected Problems 307(66)
Appendix 373(6)
Bibliography 379(2)
Index 381
Vladimir Eiderman holds a Ph.D. from Mathematical Institute of Academy of Sciences, Armenian SSR. He is Rothrock Lecturer of Indiana University. He has been Professor, Moscow State University of Civil Engineering, Visiting Professor of University of Kentucky, University of Wisconsin-Madison, and Indiana University. Dr. Eiderman has more than 30 research publications.
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