Functions of a Complex Variable

COURSE DESCRIPTION

This course is intended to serve as a formal introduction to the theory of functions of a single complex variable, enhancing its analytical and geometrical properties with the use of the computer. Complex variables have influenced a wide range of fields in science and in mathematics: from its applications to engineering, physics and quantum field theory, to helping the development of number theory, dynamical systems and algebraic geometry. This course aims to explain some of the main results for functions of a complex variable while providing sufficient mathematical background to understand computer experiments with conformal maps and iteration of complex polynomials.

Prerequisites

Multivariable Calculus, Linear Algebra, Euclidean Geometry and some basic knowledge of complex numbers. Computer experiments will require some proficiency with Python.

COURSE GOALS

During the course, students will:

  • acquire mathematical formality and develop their skills in proof writing;
  • learn how to state the main differences between a differentiable function in two real variables and a holomorphic function and state the connection among power series expansions, a (complex) analytic function and a meromorphic function;
  • develop the theory of contour integration, state and prove Cauchy’s Theorem and apply it to compute integrals of real and complex variables;
  • gain geometrical intuition behind conformal maps and normal families.
COURSE CONTENT
  1. Complex plane and elementary functions (1 week)

Complex numbers, representations of complex numbers, stereographic projection, exponential and logarithm, powers and roots, phase visualization.

  1. Complex derivative (3 weeks)

Derivative of a complex function, Cauchy-Riemann equations and holomorphic functions, inverse mapping theorem and the Jacobian, harmonic functions and conjugates, conformal maps and fractional linear transformations. Computer experiments.

  1. Complex integration (4 weeks)

Multi-calculus review, line integrals and Green’s Theorem. Fundamental Theorem of Calculus for holomorphic functions, Cauchy Theorem, Cauchy Integration Formula, Liouville’s Theorem, Morera’s Theorem and maximum modulus principle.

  1. Power series and residues (4 weeks)

Sequences and series of functions. Power series and analytic functions. Power series as holomorphic functions. Meromorphic functions, Laurent series and isolated singularities. Residue Theorem and computation of integrals using residues.

  1. Normal families and iteration of polynomials (2 weeks)

Uniform and normal convergence of sequences. Fatou and Julia sets for iterated polynomials. The Mandelbrot set and computer experiments.

Bibliography

  1. Gamelin, Theodore W. (2001). Complex analysis, Undergraduate texts in mathematics, Springer.
  2. Wergert, Elias (2012). Visual complex functions, Birkhauser.
  3. Freitag, Eberhard; Busam, Rolf (2009). Complex analysis, Universitext, Springer-Verlag.
  4. Marsden, Jerrold E.; Hoffman, Michael J. (1998). Basic complex analysis, W. H. Freeman & Co.

Support Sessions

Two hours per week with a teaching assistant

Grading

Two midterm exams (30% each) and weekly homework assignments from which, the best ten assignments will be considered to obtain the remaining 40% of your final grade