Math 352: Introduction to Complex Analysis
Introduction to Complex Analysis.
(Credit Hours:Lecture Hours:Lab Hours)
Complex algebra, analytic functions, integration in the complex plane, infinite series, theory of residues, conformal mapping.
Desired Learning Outcomes
This course is aimed at undergraduates majoring in mathematical and physical sciences and engineering. In addition to being an important branch of mathematics in its own right, complex analysis is an important tool for differential equations (ordinary and partial), algebraic geometry and number theory. Thus it is a core requirement for all mathematics majors. It contributes to all the expected learning outcomes of the Mathematics BS (see ).
Students are expected to have completed and mastered Math 290, and to have taken or to have concurrent enrollment in Math 341 (Theory of Analysis) to provide the necessary understanding of the modes of thought of mathematical analysis.
Minimal learning outcomes
Students should achieve mastery of the topics listed below. This means that they should know all relevant definitions, the full statements of the major theorems, and examples of the various concepts. Further, students should be able to solve non-trivial problems related to these concepts, and prove simple theorems in analogy to proofs given by the instructor.
- Complex numbers, moduli, exponential form, arguments of products and quotients, roots of complex numbers, regions in the complex plane.
- Limits, including those involving the point at infinity. Open, closed and connected sets. Continuity, derivatives.
- Analytic functions, Cauchy-Riemann equations, harmonic functions, finding the harmonic conjugate.
- Elementary functions in the complex plane: exponential and log functions, complex exponents, trigonometric and hyperbolic functions and their inverses.
- Contour integrals, upper bounds for moduli, primitives, Cauchy-Goursat theorem, Cauchy integral formulae, Liouville theorem, maximum modulus theorem.
- Taylor series, Laurent series, integration and differentiation of power series, uniqueness of series representation, multiplication and division of power series.
- Isolated singularities, behavior near a singularity. Residue theorem, its application to improper integrals, Jordan's lemma. Argument principle, Rouche's theorem.
- Conformal mappings. Moebius transformations.
Possible textbooks for this course include (but are not limited to):
These are at the instructor's discretion as time allows. Possibilities include applications of complex analysis in physics.