**Math 5037 – Introduction to Complex Analysis**

*Three credits. Three hours of lecture per week. Prerequisite: MATH 3153. *

Algebra of complex numbers, analytic functions, integration,

meromorphic functions, residue calculus, and conformal mappings.

## Objectives

Click here to read the course objectives.

## Syllabus

- Introduction to the Complex Numbers
- Definition of a complex numbers.
- Algebraic properties of the complex numbers.
- The roots of unity.
- The Cauchy-Schwarz and triangle inequalities.

- Functions of a complex variable
- Polynomials.
- Rational functions.
- Elementary transcendental functions.

- Open and closed sets in the complex plane
- Continuous curves and path connectedness.
- Domains.
- Continuity.
- Differentiability.
- The Cauchy-Riemann equations.

- Complex power series.
- Radius of convergence.
- Properties of power series.
- Differentiation within circle of convergence.

- Cauchy’s Theorem
- Piecewise continuously differentiable curves.
- Contour integrals.
- Antiderivatives.
- The antiderivative theorem.
- Cauchy’s Theorem.

- Cauchy integral formulae.
- Cauchy’s Formula.
- Power series and analycity.
- Holomorphicity implies analycity.
- Morera’s Theorem.
- Liouville’s theorem.
- Fundamental theorem of algebra.

- Residue theorem.
- Laurent series.
- Isolated singularities.
- Poles, removable and

essential singularities. - Meromorphic functions.
- The residue theorem.
- Maximum modulus principle.
- Rouche’s theorem, principle of the

argument. - Applications to definite integrals, summation of

series and the location of zeros.

### Bibliography

- H.A. Priestley,
*Introduction to Complex Analysis*, OUP, 1990. - I.N. Stewart and D. Tall,
*Complex Analysis – the Hitchhiker’s Guide to the Plane*, CUP, 1983. - M.R. Spiegel,
*Theory and Problems of Complex Variable*(Schaum Outline Series), McGraw-Hill, 1974.