Math 5037 Complex Analysis

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

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Syllabus

1. Introduction to the Complex Numbers
   1. Definition of a complex numbers.
   2. Algebraic properties of the complex numbers.
   3. The roots of unity.
   4. The Cauchy-Schwarz and triangle inequalities.
2. Functions of a complex variable
   1. Polynomials.
   2. Rational functions.
   3. Elementary transcendental functions.
3. Open and closed sets in the complex plane
   1. Continuous curves and path connectedness.
   2. Domains.
   3. Continuity.
   4. Differentiability.
   5. The Cauchy-Riemann equations.
4. Complex power series.
   1. Radius of convergence.
   2. Properties of power series.
   3. Differentiation within circle of convergence.
5. Cauchy’s Theorem
   1. Piecewise continuously differentiable curves.
   2. Contour integrals.
   3. Antiderivatives.
   4. The antiderivative theorem.
   5. Cauchy’s Theorem.
6. Cauchy integral formulae.
   1. Cauchy’s Formula.
   2. Power series and analycity.
   3. Holomorphicity implies analycity.
   4. Morera’s Theorem.
   5. Liouville’s theorem.
   6. Fundamental theorem of algebra.
7. Residue theorem.
   1. Laurent series.
   2. Isolated singularities.
   3. Poles, removable and
      essential singularities.
   4. Meromorphic functions.
   5. The residue theorem.
   6. Maximum modulus principle.
   7. Rouche’s theorem, principle of the
      argument.
   8. Applications to definite integrals, summation of
      series and the location of zeros.

Bibliography

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