复变函数

Chapter Ⅰ Complex Number System

1．1 Fundamental Concepts

1．2 Operations On Complex Numbers

1．3 Laws of Operations and Identities

1．4 Other Properties

1．5 Modulus of Complex Numbers

1．6 Conjugate Complex Number

1．7 Polar and Exponential Form

1．8 Operations in Exponential and Rectangular Form

1．9 Geometric Meaning of Complex Roots

1．10 Point Sets and Regions on the Complex Plane

1．11 A Brief History of Complex Numbers

1．12 Complex Number Issues on Biological Problems

Chapter Ⅱ Analytic Functions

2．1 The Concept of Complex Function

2．2 Transformations

2．3 Complex Exponential Transformations

2．4 The Limitation of Complex Numbers

2．5 Some Limitation Theorems

2．6 Limits on the Extended Complex Plane

2．7 Some Properties on Continuous Functions

2．8 Derivatives of Functions of a Complex Variable

2．9 Some Basic Differentiation Formulas

2．10 Cauchy-Riemann Conditions

2．11 Some Sufficient Conditions of Differentiability

2．12 Differentiation in Polar Coordinates

2．13 Concept and Properties of Analytic Functions

2．14 Concept and Properties of Harmonic Functions

2．15 Further Properties of Analytic Functions

2．16 Schwarz's Symmetric Principle

2．17 Analytic Functions on Some Biological Movements

Chapter Ⅲ Complex Elementary Functions

3．1 Concept and Properties of Complex Exponential Function

3．2 Concept and Properties of Complex Logarithmic Function

3．3 Complex Logarithms' s Branches and Derivatives

3．4 List of Logarithmic Identities

3．5 Concept and Properties of Complex Power Functions

3．6 Cncept and Properties of Complex Trigonometric Functions

3．7 Concept and Properties of Complex Hyperbolic Functions

3．8 An Introduction to Inverse Trigonometric and Hyperbolic Functions

3．9 Complex Exponential Functions on Transportation of Oxygen （02） and

Carbon dioxide （ CO2 ）

Chapter Ⅳ Complex Integrals

4．1 Single Variable Complex Functions with Parameters

4．2 The Concept of Definite Integrals

4．3 Classes of Curves

4．4 Contour Integrals of Functions of a Complex Variable

4．5 Numerically Evaluate Complex Integrals

4．6 Complex Antiderivatives

4．7 The Fundamental Cauchy-Goursat Theorem

4．8 On Connected Domains

4．9 Applications of Cauchy Integral Formula

4．10 Morea's Theorem ．"

4．11 Some Important Theorems

4．12 Applications of the Maximum Modulus Principle

4．13 Derivatives of Complex Functions on Hormonal Functions of

Placenta

Chapter V Complex Series

5．1 Complex Sequences

5．2 Series of Complex Numbers

5．3 An Introduction to Taylor Series

5．4 An Introduction to Laurent Series

5．5 An Introduction to Absolute and Uniform Convergence of

Power Series

5．6 Power Series and Continuous Functions

5．7 Power Series' s Integration and Differentiation

5．8 On the Representations of Complex Series

5．9 Operations On Power Series

5．10 Complex Series on Nucleoside TransplaCental Movement

Chapter VI Residues

6．1 Isolated Singularities and Residues

6．2 Residue Theorem

6．3 An Important Theorem on the Single Residue

6．4 Different Types of Isolated Singular Points

6．5 Efficient Determination of the Residues at the Various Poles

6．6 A Source of Poles

6．7 Imagining Poles and Zeros

6．8 Some Results Near Isolated Singular Points

Chapter V]I Applications of Residues

7．1 Applications of Residues in Evaluating Improper Integrals

7．2 An Application of the Fourier integral

7．3 An Important Lemma

7．4 Contour Integrations on Indented Paths

7．5 An Example of Indentation Around a Branch Point

7．6 A Special Integration

7．7 Applications of Residues in Evaluating Definite Integrals

7．8 Cauchy's Argument Principle

7．9 Consequences of Rouche' s Theorem

7．10 Applications of Inverse Laplace Transforms

Chapter Ⅵ The Geometric Interpretation of Some Analytic Fuetions

8．1 Linear Transformations

8．2 The Transformation w = 1/z

8．3 Transformations by 1/z

8．4 Fractional Linear Transformations

8．5 Cross Ratio

8．6 Transformations of the Upper Half Plane

8．7 The Transformation w = sinz

8．8 Mappings by z2 and Branches of z1/2

8．9 Square Roots of Polynomials

8．10 Riemann Surfaces

8．11 Surfaces for Related Functions

Chapter Ⅸ Conformal Mappings

9．1 Preservation of Angles

9．2 Scale Factors

9．3 Local Inverses

9．4 Harmonic Conjugates

9．5 Transformations of Harmonic Functions

9．6 Transformations of Boundary Condition

9．7 Graphing on the Complex Plane

References

Acknowledgements

《复变函数》用丰富的图例展示各种概念、定理和证明思路，以便读者理解，并能充分揭示复变函数的数学之美；《复变函数》的每一章都以一句数学家的名言开始，希望这些先辈的名言能给读者以鼓励和鞭策；在每一章节中，作者适当地编写了一些练习题，这些练习题是《复变函数》正文的延伸。作者试图让读者通过对这些问题的理解与解答，*好地掌握复变函数的基本概念和一些基本技巧。