1. GENERAL INFORMATION ABOUT NUMBERS IMPRINT AND COMPLEX NUMBERS
1.1. History of Complex Numbers
1.2. Definitions of Imaginary Numbers and Complex Numbers
1.3. Algebraic Operations Defined on the Set of Complex Numbers
1.4. Geometric Description of Complex Numbers
1.5. Polar Form of Complex Numbers
1.6. Euler's Formula, De Moivre's Identity
1.7. Roots of Complex Numbers of Various Orders
1.8. Calculation of Roots of Polynomials (Polynomials)
1.8.1. Fundamental Theorem of Algebra
1.9. Generalization of Complex Variables to Three-Dimensional Space: Quaternion

2. FUNCTIONS OF COMPLEX VARIABLES
2.1. Analysis of Various Complex Functions
2.1. 1. Polynomial Functions
2.1. 2. Rational Algebraic Functions
2.1. 3. Exponential Functions
2.1. 4. Logarithmic Functions
2.1. 5. Trigonometric Functions
2.1. 5. 1. Description of the cosz and sinz Functions
2.1. 5. 2. Description of the tanz Function
2.1. 6. Complex Hyperbolic Functions
2.1. 6. 1. Description of the coshz and sinhz Functions
2.1. 7. Inverse Trigonometric Functions
2.1. 8. Inverse Hyperbolic Functions
2.1. 9. Multi-valued Functions of the Form za and az
2.1. 9. 1. Multi-valued Functions of the Form (z – c)a
2.1. 9. 2. Definitions of Multiple-Valued Functions with Less Form
2.2. Algebraic and Transcendental Function Definitions Simple Functions
2.3. Branch Points and Branch Sections Collective Information on Riemann Surfaces
2.3.1. Branch Points and Branch Sections of the Function w = z
2.3.2. Branch Points and Branch Sections of the Function w = z–a
2.3.3. Branch Points of the Function w = n (z–a)m
2.3.4. Branch Points of the Function w = (z–a)(z–b)
2.3.5. Branch Points of the Function w = (z–a)(z–b)(z–c)
2.3.6. Branch Points and Branch Sections of the Root of Degree n of the Polynomial P(z) with Simple and Multiple Roots

3. DERIVATIVES OF COMPLEX FUNCTIONS AND CAUCHY-RIEMANN EQUATIONS DERIVATIVE OPERATORS
3.1. Derivatives of Complex Functions and Cauchy-Riemann Conditions
3.2. Harmonic Functions and Laplace Equation
3.3. Derivative Rules
3.4. Formal Similarity of Derivatives of Multivalued Functions
3.5. Derivatives of Some Functions Important in Application
3.6. Entire Function and Meromorrque Function
3.7. Derivative Operators of Complex Functions
3.7.1. Gradient Operators ∇(del) and –∇(del line)
3.7.2. Divergence of a Vector
3.7.3. Curl of a Vector
3.7.4. Laplace Operator

4. INTEGRALS OF COMPLEX FUNCTIONS
4.1. Integrals of Complex Functions Along Lines
4.1. 1. Curve Forms in the Plane
4.1. 2. Regions
4.1. 3. Complex Integration
4.1. 4. General Properties of Complex Integrals
4.1. 5. Real Components of Complex Integrals
4.1. 6. Complex Integrals Depending on Parameters
4.1. 7. Antiderivative, Antiderivative Curve of Many-Valued Functions
4.1. 7.1. Calculation of Antiderivatives for C Consisting of Finite Number of Regular Ck(k = 1, 2, ..., n–1) Curves Piece by Piece
4.1. 7.2. Green's Theorem, Cauchy's Theorem and Cauchy-Goursat Theorem in the Plane . 162
4.1. 7.3. Cauchy-Goursat Theorem in n-fold Dependent Regions
4.1. 7.4. Deformation and Shifting of Integral Lines
4.1. 7.5. Morera's Theorem . 168
4.2. Cauchy-Integral Formula and Related Formulas . 168
4.2. 1. Cuchy Integral Formula for Regions Containing More Than One Pole Point
4.2. 2. Results of Cauchy's Formula . 172
4.2. 2.1. Locating Singularity Points on the Circumference Curve
4.2. 2. Cauchy Integral for n-fold Dependent Regions
4.2. 2.3. Successive Derivatives of Cauchy's Integral Formula
4.2. 2.4. Cauchy's Inequality
4.2.2.5. Gauss's Mean Value Theorem
4.2.2.6. Liouville's Theorem
4.2.2.7. Maximum Module Theorem
4.2.2.8. Minimum Module Theorem
4.2.2.9. Angular Variation TheoremLogarithmic Residue
4.2.2.10. Rouché's Theorem
4.2.2.11. Poisson's Integral Formulas


5. COMPLEX NUMBER SEQUENCES AND SERIES Absolutes and singularities of complex functions
5.1. Complex number sequences
5.2. Complex number series
5.2.1. Convergence of complex number series
5.2.2. Criteria for convergence of complex number series
5.3. Taylor series
5.3.1. Maclaurin series
5.3.2. Taylor series of rational functions
5.3.3. Addition, conjugation, real and imaginary parts of series
5.4. Cauchy product of series
5.4.1. Multiplicative inversion of power series
5.4.2. Rearrangement of power series
5.4.3. Nesting one series into another
5.4.4. Weierstrass double series theorem
5.5. Laurent Series
5.6. Zero Points and Singular Points of Complex Functions
5.6.1. Zero Points and Zero Point Orders of Complex Functions
5.6.2. Singularity and Discrete Singular Points of Complex Functions

6. RESIDUE THEOREM AND ITS APPLICATIONS
6.1. Cauchy Residue Theorem
6.2. Methods Used in Calculating Residues
6.2.1. Calculating Residues by Integration
6.2.2. Calculating Residues in the Case of a Function Having a Simple Pole at Point z0
6.2.3. Calculating Residues of Functions Having a Higher Order Pole
6.2.4. Other Methods in Calculating Residues
6.2.4.1. Calculation of Residue of Function of Form f(z) = ϕ(z)/ψ(z) in Case ψ(z0) = 0 and ψ′(z0) ≠ 0
6.2.4.2. Calculation of Residue of Higher Functions in Complex Structure
6.2.4.3. Definite Integrals of Trigonometric Functions in [0, 2π)
6.3. Improper Integrals
6.3.1. Infinite Region Integrals
6.3.2. Infinite Region Integrals
6.4. Auxiliary Theorems Used in Calculation of Infinite Integrals
6.4.1. First Jordan Auxiliary Theorems
6.4.2. Second Jordan Auxiliary Theorem
6.4.2.1. Second Jordan Lemma for Functions with Exponential Term Factors
6.5. Calculation of Residue at Infinity
6.5.1. Two Finite Regions and Calculation of Residue at Infinity
6.6. Examples of Integrals of Meromorc Functions
6.7. Plemelj-Sokhostki Prime Value Integral and Prime Value Integral Involving a Semicircular Arc
6.7.1. Plemelj-Sokhostki Prime Value Integral
6.7.2. Prime Value Formulas Involving a Semicircular Arc
6.8. Integral of Functions with Branch Points
6.9. Integral of Functions with More Than One Branch Point in a Finite Region, Other Indirect Integrals
6.10. Addition of Series Using the Residue Theorem
6.11. Mittag-Lefsher Expansion Theorem

7. ANALYTIC CONTINUATION, INFINITE PRODUCTION SERIES
7.1. Analytical Continuation
7.1.1. Definitions and Basic Concepts
7.1.2. Analytical Continuation by Expansion Method for Taylor Series
7.1.3. Natural Limit or Barrier of Analytical Continuation
7.1.4. Riemann Theorem and Schwartz Reflection Principle on Analytical Continuation

8. OPEN CONSTRUCTIVE DESCRIPTIONS
8.1. Basic Properties of Open Conformal Descriptive Descriptive
8.2. Open Conformal Transformations of Small Shapes
8.3. Transformation in the Case Where the Derivative of the Descriptive Function is Zero of Higher Order
8.4. Riemann Descriptive Theorem .428
8.5. Some Basic Transformations .430
8.5.1. Translation, Rotation, Magnification and Convolution Transformations .430
8.5.2. Bilinear Transformation .432
8.5.3. Jaukovsky Transformation .435
8.5.4. Schwarz-Christoffel Transformation
8.5.4a. Information for Preserving the Schwarz-Christoffel Transformation
8.5.4b. Schwarz-Christoffel Transformation


9. INTRODUCTION TO SINGULAR INTEGRATED EQUATIONS
9.1. Classification of Integral Equations
9.1. 1. Fredholm Integral Equations
9.1. 2. Volterra Integral Equations
9.1. 3. Singular Integral Equations
9.2. Fundamentals of Singular Integral Equations
9.2. 1. Integral Lines of Singular Integral Equations
9.2. 2. Topics on Functions Entering Singular Integral Equations Hölder-Lipschitz (H-L) (Hα) Conditions
9.2. 3. Integrals of Cauchy Type . 462
9.2. 3a. Cauchy Formulas Defined on Simple, Piecewise, Regular, Continuous and Closed Lines (Appendix to Section 4.2)
9.2. 3b. Cauchy Formulas for Special Density Functions
9.2.3c. Cauchy Type Integral Calculated on Non-Self-Intersecting (Simple), Piecewise, Regular, Open or Closed Integration Curves
9.2.4. Unique Integrals, Principal Values ​​of Cauchy Type Integrals
9.2.5. Sokhotski-Plemelj Formulas
9.2.5.1. Sokhotski-Plemelj Formulas for Angular Integration Lines7
9.2.6. Cauchy Type Integral Calculated on Infinite Straight Lines
9.2.7. Calculation of Cauchy Type Integral Given on Infinite Straight Lines by Residue Theorem
9.2.8. Conditions for the Given ϕ(t) Function on Non-Self-Intersecting (Simple), Regular and Closed L Lines to be the Boundary Value of the Analytic Function in the Interior Region and to be the Boundary Value of the Analytic Function in the Exterior Region
9.2.9. Changing the Order of Successive Integrations Poincaré-Bertrand Formula
9.2.9.1. Poincaré-Bertrand Formula with a Single Variable Density Function and Calculation of Cauchy Singular Integral Using This Formula
9.2.10. Harnack's Theorem
9.2.11. Extension of Integration Line to (n + 1)-Fold Connected Regions
9.2.12. Piecewise Analytic Function for the Set of Closed Loop Curves L
9.2.13. Relation Between Cauchy Kernel and Other Strong Singular Kernels
9.2.13.1. Relation Between Hilbert Kernel and Cauchy Kernel
9.2.13.2. Relation Between Schwarz Kernel and Hilbert Kernel and Cauchy Kernel in the Limit Case
9.2.14. Behavior of Cauchy Type Integration at Ends of Integration Line and Discontinuity Points of Density Function
9.2.14.1. Asymptotic Expansions of Singularity at Ends a and b of Open L Integration Line
9.2.14.2. Case of Density Function Showing Finite Jump Discontinuity
9.2.14.3. Case of Density Function Having Force Singularity at Ends and on Integration Arc
9.2.15. Conversion of Singular Integration with Logarithmic Kernel to Singular Integration with Cauchy Kernel

9.3.7. Exceptional (Discrete) Cases of RHP
9.3.7.1. Solution of RHP when the Optimal Function G(t) Goes from Integer Order to Zero and Infinity at Some Points on the Closed Boundary Curve
9.3.7.2. Case of RHP when the Closed Boundary Line L is Angular
9.4. Solution of Singular Integral Equations with Cauchy and Hilbert Kernels in the Case of Closed Boundary Lines by Transforming them into RHP
9.4.1. Solution of Singular Integral Equations with Cauchy Kernels
9.4.2. Solution of Singular Integral Equations with Hilbert Kernels
9.4.3. Approximate Solution of Singular Integral Equations with Closed Integration Lines
9.4.4. Solution of Singular Integral Equations in the Case of Closed Integration Lines
9.4.5. Solution of Singular Integral Equations by Reduction to Regular Integral Equations (Fredholm type) by the Regularization Method
9.4.5.1. Some Properties of Singular Operators
9.4.5.2. Regularization Operations, Conjugate Operations
9.4.5.3. Solution of Singular Integral Equations by the Carlemann–Vekua Method
9.5. Riemann-Hilbert Boundary Value Problems with Open Boundary Lines and Open Ends Converging at Nodal Points
9.5.1. Behavior of Cauchy Type Integral at Integration Line Ends and Density Function Discontinuities
9.5.2. Cauchy Type Integral with Density Function Discontinuity of the First Kind
9.5.3. Asymptotic Values ​​of Cauchy Type Integral in the Case of Exponential Branch Singularity of the Density Function
9.5.4. Asymptotic Behavior of Cauchy Type Integral and Principal Value Integral at Nodal Points Where Ends of Open Boundary Lines Meet
9.5.5. Solution of Riemann-Hilbert Problem (RHP) Consisting of Regular Open Boundary Lines (Arcs)
9.5.5.1. Solution of RHP in the Case of Single Open Boundary Line
9.5.5.2. Solution of RHP in the Case of n Open and Non-Intersecting Regular Boundary Lines
9.5.5.3. Solution of RHP with Open Boundary Lines with Ends Meet at Nodal Points
9.5.5.4. Closed Solution of Dominant Singular Integral Cycle
9.5.6. Introduction to Numerical Solution Methods of Singular Integral Equations with Open Boundary Lines
9.5.6.1. Jacobi Polynomial Information to be Used in the Solution
9.5.6.2. Solution of Integral Equations with Cauchy Kernel of the First Kind with Jacobi Polynomials
9.5.6.3. Solution of Integral Equation with Cauchy Kernel of the Second Kind