Topics in Complex Analysis
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Cauchy-Green's Formula and relation to Cauchy's Integral formula. Power series expansion of holomoprhic functions. Cauchy-Hadamard formula. Theorem of Morera.
Complex numbers | Algebra (all content) | Math | Khan Academy
Cauchy's estimates. Liouville's theorem. Application to the Fundamental Theorem of Algebra.
Compact convergence and Weierstrass theorem. Order of zeroes. The identity principle. Laurent series expansion in a ring. Isolated singularities.
Introduction to Complex Analysis
Removable singularities, poles, essential singularities. Riemann extension theorem. Hours - Recitation:.
Hours - Total Credit:. Typical Scheduling:.
Topics from complex function theory, including contour integration and conformal mapping. Course Text:.
Topic Outline:. This course is designed to introduce the student to the basic results in complex variable theory, in particular Cauchy's theorem, and to develop the student's facility in the following three areas: Computing the Laurent or Taylor series expansions associated to a function which is analytic in part of the complex plane, and the determination of the region of convergence of such series Computing definite integrals by means of the residue calculus Solving boundary value problems associated to the Laplace operator by means of the conformal transformations associated to analytic functions Topics to be covered: Geometry of the complex plane, triangle inequalities, geometric proof of the fundamental theorem of algebra Analytic functions: Continuity and differentiability, the Cauchy-Riemann equations.
Complex functions as maps of the complex plane into itself Elementary analytic functions, including the logarithm, and its principle branch, log z Line integrals, the Cauchy integral formula and the Cauchy-Goursat theorem.
Proof of the Cauchy formula to be based on Green's theorem.