The main purpose of esta m Second Edition is essentially the same as the primer edition with changes designated a Continuation.

“The theory of functions of one variable complex variables Also called complex of brevity or Complex Analysis, it is one of the most beautiful branches, as well as useful mathematics. Despite its origins in an atmosphere of mystery, suspicion and distrust, as evidenced by the imaginary and complex terms in the literature, was finally placed on a solid base line 19th century through the efforts of Cauchy, Riemann, Weierstrass, Gauss and other mathematical Great “.

This book esta Designed for use as a supplement to all current texts or standards As textbook COURSE FOR A formal theory of complex variable AND APPLICATIONS. Also be of great value to those who take courses in mathematics, physics, aerodynamics, elasticity, and many other fields of science and engineering.

Numerous problems theorems and derivations of formulas INCLUDED also on the problems solved. Each chapter begins with a clear statement of relevant nitions, principles and theorems together with illustrative material Description and Other. This is followed by sets of Graduate and supplementary Problems Resolved. The large number of additional problems with answers serve for a more complete view of each chapter materials.

Topics covered include algebra and geometry of complex numbers, calculus differential and integral Complex in finite series, including Taylor and Laurent series, the theory of waste with Applications to s evaluation of integrals and series, and Representation conforme with applications from various fields.

Some adj Of changes we made to the first edition son the Following:

(A) We have expanded and corrected many of Sections sea for more accessible Our readers para.

(B) We have changed the format of the text, for example, the number of chapters, are now included on the label of all sectors, examples and problems.

(C) Results Many formally expressed as propositions and theorems.

1.1 The Real Number System

1.2 Graphical Representation of Real Numbers

1.3 The Complex Number System

1.4 Fundamental Operations with Complex Numbers

1.5 Absolute Value

1.6 Axiomatic Foundation of the Complex Number System

1.7 Graphical Representation of Complex Numbers

1.8 Polar Form of Complex Numbers

1.9 De Moivre’s Theorem

1.10 Roots of Complex Numbers

1.1 Euler’s Formula

1.12 Polynomial Equations

1.13 The nth Roots of Unity

1.14 Vector Interpretation of Complex Numbers

1.15 Stereographic Projection

1.16 Dot and Cross Product

1.17 Complex Conjugate Coordinates

1.18 Point Sets

CHAPTER 2 FUNCTIONS, LIMITS, AND CONTINUITY 41

2.1 Variables and Functions

2.2 Single and Multiple-Valued Functions

2.3 Inverse Functions

2.4 Transformations

2.5 Curvilinear Coordinates

2.6 The Elementary Functions

2.7 Branch Points and Branch Lines

2.8 Riemann Surfaces

2.9 Limits

2.10 Theorems on Limits

2.1 Inﬁnity

2.12 Continuity

2.13 Theorems on Continuity

2.14 Uniform Continuity

2.15 Sequences

2.16 Limit of a Sequence

2.17 Theorems on Limits of Sequences

2.18 Inﬁnite Series

CHAPTER 3 COMPLEX DIFFERENTIATION AND THE CAUCHY–RIEMANN EQUATIONS 7

3.1 Derivatives

3.2 Analytic Functions

3.3 Cauchy–Riemann Equations

3.4 Harmonic Functions

3.5 Geometric Interpretation of the Derivative

3.6 Differentials

3.7 Rules for Differentiation

3.8 Derivatives of Elementary Functions

3.9 Higher Order Derivatives

3.10 L’Hospital’s Rule

3.1 Singular Points

3.12 Orthogonal Families

3.13 Curves

3.14 Applications to Geometry and Mechanics

3.15 Complex Differential Operators

3.16 Gradient, Divergence, Curl, and Laplacian

CHAPTER 4 COMPLEX INTEGRATION AND CAUCHY’S THEOREM 1

4.1 Complex Line Integrals

4.2 Real Line Integrals

4.3 Connection Between Real and Complex Line Integrals

4.4 Properties of Integrals

4.5 Change of Variables

4.6 Simply and Multiply Connected Regions

4.7 Jordan Curve Theorem

4.8 Convention Regarding Traversal of a Closed Path

4.9 Green’s Theorem in the Plane

4.10 Complex Form of Green’s Theorem

4.1 Cauchy’s Theorem. The Cauchy–Goursat Theorem

4.12 Morera’s Theorem

4.13 Indeﬁnite Integrals

4.14 Integrals of Special Functions

4.15 Some Consequences of Cauchy’s Theorem vii

CHAPTER 5 CAUCHY’S INTEGRAL FORMULAS AND RELATED THEOREMS 144

5.1 Cauchy’s Integral Formulas

5.2 Some Important Theorems

CHAPTER 6 INFINITE SERIES TAYLOR’S AND LAURENT’S SERIES 169

6.1 Sequences of Functions

6.2 Series of Functions

6.3 Absolute Convergence

6.4 Uniform Convergence of Sequences and Series

6.5 Power Series

6.6 Some Important Theorems

6.7 Taylor’s Theorem

6.8 Some Special Series

6.9 Laurent’s Theorem

6.10 Classiﬁcation of Singularities

6.1 Entire Functions

6.12 Meromorphic Functions

6.13 Lagrange’s Expansion

6.14 Analytic Continuation

CHAPTER 7 THE RESIDUE THEOREM EVALUATION OF INTEGRALS AND SERIES 205

7.1 Residues

7.2 Calculation of Residues

7.3 The Residue Theorem

7.4 Evaluation of Deﬁnite Integrals

7.5 Special Theorems Used in Evaluating Integrals

7.6 The Cauchy Principal Value of Integrals

7.7 Differentiation Under the Integral Sign. Leibnitz’s Rule

7.8 Summation of Series 7.9 Mittag–Lefﬂer’s Expansion Theorem

7.10 Some Special Expansions

CHAPTER 8 CONFORMAL MAPPING 242

8.1 Transformations or Mappings

8.2 Jacobian of a Transformation

8.3 Complex Mapping Functions

8.4 Conformal Mapping

8.5 Riemann’s Mapping Theorem

8.6 Fixed or Invariant Points of a Transformation

8.7 Some General Transformations

8.8 Successive Transformations

8.9 The Linear Transformation

8.10 The Bilinear or Fractional Transformation

8.1 Mapping of a Half Plane onto a Circle

8.12 The Schwarz–Christoffel Transformation

8.13 Transformations of Boundaries in Parametric Form

8.14 Some Special Mappings

CHAPTER 9 PHYSICAL APPLICATIONS OF CONFORMAL MAPPING 280

9.1 Boundary Value Problems

9.2 Harmonic and Conjugate Functions

9.3 Dirichlet and Neumann Problems

9.4 The Dirichlet Problem for the Unit Circle. Poisson’s Formula

9.5 The Dirichlet Problem for the Half Plane

9.6 Solutions to Dirichlet and Neumann Problems by Conformal Mapping Applications to Fluid Flow

9.7 Basic Assumptions 9.8 The Complex Potential

9.9 Equipotential Lines and Streamlines

9.10 Sources and Sinks

9.1 Some Special Flows

9.12 Flow Around Obstacles

9.13 Bernoulli’s Theorem

9.14 Theorems of Blasius Applications to Electrostatics

9.15 Coulomb’s Law

9.16 Electric Field Intensity. Electrostatic Potential

9.17 Gauss’ Theorem

9.18 The Complex Electrostatic Potential

9.19 Line Charges

9.20 Conductors

9.21 Capacitance Applications to Heat Flow

9.2 Heat Flux

9.23 The Complex Temperature

Schaum / Lipschutz/ Murray R. Spiegel/ Schiller & Spellman

2nd Edition

978-0-07-161569--3 | 978-0-07-161570-9

eBook

English

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