#### Description

This course is an introduction to complex variables accessible to juniors and seniors in engineering, physics and mathematics. The algebra of complex numbers, analytic functions, Cauchy Integral Formula, theory of residues and application to the evaluation of real integrals, conformal mapping and applications to physical problems.

#### Prerequisites

MATH 241 or MATH 340

#### Topics

**Complex Numbers**

Complex arithmetic

Geometric representation

Polar form

Powers

Roots

Elementary plane topology

**Analytic Functions**

Continuity

Differentiability

Cauchy-Riemann equations

Analytic functions

Harmonic functions and harmonic conjugates

**Contour Integrals**

Upper bound estimates

Anti-derivatives

Cauchy-Goursat theorem

Cauchy integral formulas

Liouville's theorem

Fundamental theorem of algebra

Maximum modulus theorem

**Elementary functions**

Exponential function

Logarithmic function

Trigonometric functions

Hyperbolic functions

The functions zc and cz

**Infinite sequences and series**

Sequences and series of constants

Sequence and series of functions

Geometric series

Power series and Taylor series

Laurent series

**Residues**

Isolated singularities

Resides and the residue theorem

Evaluation of real integrals by residues

**Boundary value problems and applications**

Conformal mappings

Mapping properties of some elementary functions

Application to the steady state heat flow and electrostatic potential