**TDP**

This short tutorial is a companion material to the course on Functions of a Complex Variables . It is intended to help the student, but will replace neither personal lecture notes nor a good textbook.

From time to time, Maple commands are added in examples.

- Contents
- Complex numbers.
- Algebraic form.
- Geometric representation of complex numbers.
- Polar form.
- De Moivre's formula.
- Roots of a complex number.
- Equations of degree 2.
- Polynomial equations of higher degree.

- Functions of a complex variable.

- Local properties of a function.
- Limits and continuity.
- Derivation.
- Cauchy-Riemann Equations.
- Harmonic functions.
- Analytic functions.

- A catalogue of analytic functions.
- Exponential in basis .
- Trigonometric functions.
- Hyperbolic functions.
- The logarithm of a complex number.
- Analyticity of the logarithmic function.
- Complex exponentials.
- Inverse trigonometric functions.
- Inverse hyperbolic functions.

- Integrals.
- Line integral.
- Theorem LM.
- Cauchy's Theorems.
- Cauchy's Integral Formula.
- Generalization of Cauchy's Integral Formula.
- Morera's Theorem.

- The theorems of Liouville and d'Alembert.
- Maximum principle and minimum principle.
- Liouville's Theorem.
- The Fundamental Theorem of Algebra.
- Weierstrass' Theorem.

- Complex series.

- Power Series.

- Residues and Integrals.

- Conformal mappings.

- About this document ...

Noah Dana-Picard 2007-12-24