A review of complex numbers and the complex plane.

Computer-aided handling of complex numbers using Microsoft Excel.

Computer-aided handling

Absolute values

General arguments and principal argument of a complex number.

Arguments

Polar form

Meaning of complex numbers and their use.

Complex numbers

Some important properties of complex numbers.

Properties of complex numbers

The Euler formula and exponential form of complex numbers.

Euler's formula

Introduction

Various terminologies for describing sets of points in the complex plane, and their meanings.

General neighborhoods and deleted neighborhoods.

Neighborhoods

Disks

Interior, boundary, exterior, limit and isolated points.

Points

Point sets

Half-planes

Continuous, smooth, piecewise-smooth, simple, closed, and simple closed curves, contour, polygonal path or line.

Paths

Connected, convex, star-like or star-shaped sets.

Structure of point sets

Simply-connected and multiply-connected domains, open and closed regions.

Domains and regions

Point Sets in the Complex Plane

Introduction to functions of complex variables.

Meaning of function of a complex variable.

Definition

Domain of definition of a function of a complex variable.

Domain of definition

Worked examples on the domain of definition of functions of complex variables.

Worked examples I

Graphing

Functions of Complex Variables

Limits of functions of complex variables.

Informal definition of the limits of complex functions, and the conditions for their existence.

Informal definition

Formal definition of the limits of complex functions, and the conditions for their existence.

Formal definition

Worked examples on the proof of the existence of limits of complex functions.

Evaluation

Worked examples on the evaluation of limits of complex functions.

Worked examples II

More worked examples on the evaluation of limits of complex functions.

Worked examples III

Worked examples IV

Limits

Continuity of functions of complex variables.

Informal definition of the continuity of complex functions.

Formal definition of the continuity of complex functions.

Worked examples on the continuity of complex functions.

More worked examples on the continuity of complex functions.

Continuity

Differentiability of functions of complex variables.

Definition of differentiability of complex functions.

Worked examples on the differentiability of complex functions.

More worked examples on the differentiability of complex functions.

Differentiability

Necessary and sufficient conditions for the existence of the derivatives of complex functions.

Complex Variables

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