دانلود کتاب An Introduction to Complex Analysis and the Laplace Transform

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Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Preface
Author
Introduction
1. Complex Numbers and Their Arithmetic
1.1. Complex Numbers
1.2. Operations with Complex Numbers
2. Functions of a Complex Variable
2.1. The Complex Plane
2.1.1. Curves in the complex plane
2.1.2. Domains
2.2. Sequences of Complex Numbers and Their Limits
2.3. Functions of a Complex Variable; Limits and Continuity
3. Differentiation of Functions of a Complex Variable
3.1. The Derivative. Cauchy-Riemann Conditions
3.1.1. The derivative and the differential
3.1.2. Cauchy-Riemann conditions
3.1.3. Analytic functions
3.2. The Connection between Analytic and Harmonic Functions
3.3. The Geometric Meaning of the Derivative. Conformal Mappings
3.3.1. The geometric meaning of the argument of the derivative
3.3.2. The geometric meaning of the modulus of the derivative
3.3.3. Conformal mappings
4. Conformal Mappings
4.1. Linear and Mobius Transformations
4.1.1. Linear functions
4.1.2. Mobius transformations
4.2. The Power Function. The Concept of Riemann Surface
4.3. Exponential and Logarithmic Functions
4.3.1. Exponential function
4.3.2. The logarithmic function
4.4. Power, Trigonometric, and Other Functions
4.4.1. The general power function
4.4.2. The trigonometric functions
4.4.3. Inverse trig functions
4.4.4. The Zhukovsky function
4.5. General Properties of Conformal Mappings
5. Integration
5.1. Definition of the Contour Integral
5.1.1. Properties of the contour integral
5.2. Cauchy-Goursat Theorem
5.3. Indefinite Integral
5.4. The Cauchy Integral Formula
6. Series
6.1. Definitions
6.2. Function Series
6.3. Power Series
6.4. Power Series Expansion
6.5. Uniqueness Property
6.6. Analytic Continuations
6.7. Laurent Series
7. Residue Theory
7.1. Isolated Singularities
7.2. Residues
7.3. Computing Integrals with Residues
7.3.1. Integrals over closed curves
7.3.2. Real integrals of the form 2R0R(cos°; sin°) d°; where R is a rational function of cos °and sin °
7.3.3. Improper integrals
7.4. Logarithmic Residues and the Argument Principle
8. Applications
8.1. The Schwarz-Christo el Transformation
8.2. Hydrodynamics. Simply-connected Domains
8.2.1. Complex potential of a vector field
8.2.2. Simply-connected domains
8.3. Sources and Sinks. Flow around Obstacles
8.3.1. Sources and sinks
8.3.2. Vortices
8.3.3. Flow around obstacles
8.3.4. The Zhukovsky airfoils
8.3.5. Lifting force
8.4. Other Interpretations of Vector Fields
8.4.1. Electrostatics
8.4.2. Heat ow
8.4.3. Remarks on boundary value problems
9. The Laplace Transform
9.1. The Laplace Transform
9.2. Properties of the Laplace Transformation
9.3. Applications to Differential Equations
9.3.1. Linear ODEs
9.3.2. Finding the original function from its transform
9.3.3. Differential equations with piecewise defined right hand sides
9.3.4. Application of the convolution operation to solving differential equations
9.3.5. Systems of differential equations
Solutions, Hints, and Answers to Selected Problems
Appendix
Bibliography
Index