Complex Variables with Applications

Complex Numbers.

Introduction.

More Properties of Complex Numbers.

Complex Numbers and the Argand Place.

Integer and Fractional Powers of Complex Numbers.

Loci, Points, Sets and Regions in the Complex Plane.

The Complex Function and its Derivative.

Introduction.

Limits and Continuity.

The Complex Derivative.

The Derivative and Analyticity.

Harmonic Functions.

Some Physical Applications of Harmonic Functions.

The Basic Transcendental Functions.

The Exponential Function.

Trigonometric Functions.

Hyperbolic Functions.

The Logarithmic Function.

Analyticity of the Logarithmic Function.

Complex Functions.

Inverse Trigonometric and Hyperbolic Functions.

More on Branch Cuts and Branch Points.

Appendix: Phasors.

Integration in the Complex Plane.

Introduction to Line Integration.

Complex Line Integration.

Contour Integration and Green's Theorem.

Path Independence, Indefinite Integrals, Fundamental Theorem of Calculus in the Complex Plane.

The Cauchy Integral Formula and it Extension.

Some Applications of the Cauchy Integral Formula.

Introduction to Dirichlet Problems - The Poisson Integral Formula for the Circle and Half Plane.

Appendix: Green's Theorem in the Plane.

Infinite Series Involving a Complex Variable.

Introduction and Review of Real Series.

Complex Sequences and Convergence of Complex Series.

Uniform Convergence of Series.

Power Series and Taylor Series.

Techniques for Obtaining Taylor Series Expansions.

Laurent Series.

Properties of Analytic Functions Related to Taylor Series: Isolation of Zeros, Analytic Continuation, Zeta Function, Refelction.

The z transformation.

Appendix A: Fractals and the Mandelbrot Set.

Residues and Their Use in Integration.

Introduction Definition of the Residue.

Isolated Singularities.

Finding the Residue.

Evaluation of Real Inte