Material: A description of the covered topics, homework assignments and all announcements will be posted weekly on the course webpage. Office Hours: MF 11-noon, W 1-2pm; About the Course. The course Introduction to Complex Analysis explores the additional structure provided by complex differentiation. While real analysis conveys a rather pessimistic point of view, you will quickly realize.
Applying the Cauchy-Riemann Equations, Cauchy's Theorem, Cauchy's Integral Formula, Cauchy's Inequality, Liouville's Theorem and the Maximum Modulus Principle to complex valued functions. Applying Taylor's Theorem, Laurent's Theorem and the Residue Theorem. Students should be able to apply the Residue Theorem to evaluate improper integrals.
The maximum modulus principle, Schwarz’s lemma, the Riemann mapping theorem and the Weierstrass factorization theorem, Harmonic families and Poisson's formula, Green’s Function, Non-Euclidean Geometry, Riemannian metrics, curvature, additional topics as time permits and interest dictates: theorems of Runge problems, elliptic functions, zeros of analytic functions, the Schwarz-Christoffel.
It covers the homework problems in Chapter 2. The first midterm will be on Oct. 26 in regular class. It will cover everything till the proof of Cauchy Goursat Theorem on P150. The third quiz will be on Nov. 16 and 17 in your tutorial class. It covers the sections from the proof of Cauchy-Goursat theorem to maximum modulus principle.
Maximum modulus theorem, Schwarz’s Lemma. . Homework assignments will appear on this page approximately every week. Students are strongly advised to work on all the homework problems to make sure they are keeping pace with the class. The final grade will be based only on the homework grades. The learning goals for this course are to master the basic ideas and tools of complex function.
Math 448: Complex Variables (3 credits) Course Description This course is for students who desire a rigorous introduction to the theory of functions of a complex variable. Topics include Cauchy's theorem, the residue theorem, the maximum modulus theorem, Laurent series, the fundamental theorem of algebra, and the argument principle.
There will be extensive homework assignments involving a mix of theory and computational experiments, and an in-class final. Topics covered in the class include floating-point arithmetic, solving large linear systems, eigenvalue problems, interpolation and quadrature (approximation theory), nonlinear systems of equations, linear and nonlinear least squares, nonlinear optimization, and Fourier.
Theorem, the Maximum Modulus Principle, Liouville’s Theorem, harmonic functions, Taylor and Laurent series, singularities, the Residue Theorem, conformal mapping, normal families, the Riemann Mapping Theorem. Course Objectives The main objective of Complex Analysis is to study the development of functions of one complex variable. Students will perform a thorough investigation of the major.