# Math 478 Complex Analysis

Spring 2014

Complex analysis is one of the most beautiful corners of mathematics.  It studies the functions with complex number inputs and outputs that share some properties with polynomials:  zeros can be counted, function values at all points are determined by the values along a curve, power series expressions are available at every point.

Special as they are, applications in mathematics and science are legion.  The classical geometry of surfaces can be based on complex analysis.  The greatest theorems of number theory are proved using complex analysis. Special functions such as Bessel functions arise naturally in physics. Iteration of analytic functions generates fabulous fractals. Etc.

Course components

• The complex plane and linear fractional transformations
• Basic functions:  exponential, log, sine, cosine
• Cauchy’s Theorem and consequences
• Varied selection of applications
• Student project

Properties of analytic functions

Prime numbers in arithmetic progressions