Complex Analysis
(Complete Item Description)
- Abstract:
This course is an introduction to complex analysis, or the theory of the analytic functions of a complex variable. Put differently, complex analysis is the theory of the differentiation and integration of functions that depend on one complex variable. Because of the algebraic properties of the complex numbers and the inherently geometric flavor of complex analysis, this course will feel quite different from Real Analysis, although many of the same concepts, such as open sets, metrics, and limits will reappear. Upon successful completion of this course, the student will be able to: manipulate complex numbers in various representations, define fundamental topological concepts in the context of the complex plane, and define and calculate limits and derivatives of functions of a complex variable; represent analytic functions as power series on their domains and verify that they are well-defined; define a branch of the complex logarithm; classify singularities and find Laurent series for meromorphic functions; state and prove fundamental results, including Cauchy’s Theorem and Cauchy’s Integral Formula, the Fundamental Theorem of Algebra, Morera’s Theorem and Liouville’s Theorem; use them to prove related results; calculate contour integrals; calculate definite integrals on the real line using the Residue Theorem; define linear fractional transformations and prove their essential characteristics; find the image of a region under a conformal mapping; state, prove, and use the Open Mapping Theorem. This free course may be completed online at any time. (Mathematics 243)
- Subject:
- Mathematics and Statistics
- Grade Level:
- Post-secondary
- Collection:
- Saylor Foundation