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　Concept 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