Chapter 1: Complex Numbers

1.1 Complex Arithmetic

We perform arithmetic operations on complex numbers.

1.2 The Complex Plane

We learn a geometric interpretation of complex numbers.

1.3 Complex Conjugate

We learn properties of the complex conjugate.

1.4 The Complex Variable, z

We learn to use a complex variable.

1.5 Operations in the Complex Plane

We visualize operations on complex numbers in the complex plane.

1.6 Special Sets

We learn to recognize and sketch special sets in the complex plane.

1.7 Hyperbolic Functions

We learn the basic properties of the hyperbolic functions.

Chapter 2: Complex Functions

2.1 Complex Functions

We introduce functions of a complex variable.

2.2 Complex Logarithms

We define and discuss complex logarithms.

2.3 Complex Trigonometric Functions

We define and discuss the complex trigonometric functions.

2.4 Complex Mappings

We determine the image of special sets.

Chapter 3: Analytic Functions

3.1 Complex Limits

We find limits of complex functions.

3.2 Complex Derivatives

We find derivatives of complex functions

3.3 Cauchy-Riemann Equations

We derive the Cauchy-Riemann equations.

3.4 Analytic Functions

We determine if a function is analytic using the Cauchy-Riemann equations.

3.5 Harmonic Functions

We determine and create harmonic functions.

Chapter 4: Sequences and Series

4.1 Complex Sequences

We find limits of complex sequences.

4.2 Complex Series

We determine convergence of complex series.

4.3 Power Series

We determine radius of convergence of complex power series.

4.4 Taylor Series

We determine the Taylor series for a given analytic function.

4.5 Laurent Series

We determine the Laurent series for a given function.

Chapter 5: Complex Integrals

5.1 Contour Integrals

We compute integrals of complex functions along contours.

5.2 Cauchy’s Theorem

We compute integrals of complex functions around closed curves.

5.3 Residue Theorem

We use the Residue Theorem to compute integrals of complex functions around closed contours.

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