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Complex Numbers and their Geometric Representation / Section I：

Foundations / Chapter I：

Introduction / 1：

The system of real numbers / 2：

Pointgs and vectors of the plane / 3：

The System of Complex Numbers and the Gaussian Plane of Numbers / Chapter II：

Historical remarks / 4：

Introduction of complex numbers / 5：

Notation

Equality and inequality / 6：

Addition and subtraction / 7：

Multiplication and division / 8：

Derived rules / 9：

Powers

The system of complex numbers as an extension of the system of real numbers / 10：

Trigonometric representation of complex numbers / 11：

Geometric representation of multiplication and division / 12：

Inequalities and absolute values / 13：

Examples

The Riemann Sphere of Numbers / Chapter III：

The stereographic projection / 14：

The Riemann sphere of numbers / 15：

The point infinity

Linear Functions and Circular Transformations / Section II：

Mapping by Means of Linear Functions / Chapter IV：

Mapping by means of entire linear functions / 16：

Mapping by means of the function w = 1/z / 17：

Mapping by means of arbitrary linear functions / 18：

Normal Forms and Particular Linear Mappings / Chapter V：

The group-property of linear transformations / 19：

Fixed points and normal forms / 20：

Particular linear mappings / 21：

Cross ratios

Further examples / 22：

Sets and Sequences / Section III：

Power Series

Point Sets and Sets of Numbers / Chapter VI：

Point sets / 23：

Sets of real numbers / 24：

The Bolzano-Weierstrass theorem / 25：

Sequences of Numbers / Chapter VII：

Infinite Series

Sequences of complex numbers / 26：

Sequences of real numbers / 27：

Infinite series / 28：

The circle of convergence / Chapter VIII：

Operations on power series / 30：

Analytic Functions and Conformal Mapping / Section IV：

Functions of a Complex Variable / Chapter IX：

The concept of a function of a complex variable / 31：

Limits of functions / 32：

Continuity / 33：

Differentiability / 34：

Properties of functions represented by power series / 35：

Analytic functions / Chapter X：

Conformal mapping / 37：

The Elementary Functions / Section V：

Power and Root / Chapter XI：

The Rational Functions

Power and root / 38：

The entire rational functions / 39：

The fractional rational functions / 40：

The Exponential, Trigonometric, and Hyperbolic Functions / Chapter XII：

The exponential function / 41：

The functions cos z and sin z / 42：

The functions tan z and cot z / 43：

The hyperbolic functions / 44：

The Logarithm, the Cyclometric Functions, and the Binomial Series / Chapter XIII：

The logarithm / 45：

The cyclometric functions / 46：

The binomial series and the general power Bibliography / 47：

Index

Complex Numbers and their Geometric Representation / Section I：

Foundations / Chapter I：

Introduction / 1：

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