MATH-505: Complex Analysis

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This is an introductory course in complex analysis, giving the basics of the theory along with applications, with an emphasis on applications of complex analysis and especially conformal mappings. Students should have a background in real analysis (as in the course Real Analysis I), including the ability to write a simple proof in an analysis context.

• The Concept of Analytic Functions: The complex numbers and the complex plane<, Functions of a complex variable, General properties of analytic functions, Linear transformations, Basic properties of linear transformation, mapping for problems, stereographic projections, Basic concepts of conformal mapping, The exponential and the logarithmic functions, the trigonometric functions, Taylor’s series, Laurent’s series, infinite series with complex terms, power series, infinite products.
• Integration in the Complex Domain: Cauchy’s theorem, Cauchy’s integral formula and its applications, Laurent’s expansion, isolated singularities of analytic functions, the residue theorem and its applications.
• Contour Integration: Definite integrals, partial fraction, expansion of $\cot 2z$,
• The arguments principle theorem and its applications: Rouche’s theorem,
• Analytic Continuation: The principle of Analytic Continuation.

1. J.W Brown and R.V Churchill, Complex Variables and Applications, 8th Edition, McGraw-Hill, 2009.
2. Dennis Zill, A first course in complex analysis with applications, Jones and Bartlett Publishers, Inc., 2008.
3. J.H. Mathews and R.W. Howell, Complex analysis for mathematical engineering, Norosa Publishing House Dehli, 2006.

• http://en.wikipedia.org/wiki/Complex_number
• http://math.furman.edu/~dcs/courses/math39/lectures/lecture-5.pdf
• http://www.math.wisc.edu/~angenent/Free-Lecture-Notes/freecomplexnumbers.pdf
• Computational Knowledge Engine: http://www.wolframalpha.com
• http://en.wikipedia.org/wiki/Stereographic_projection
• http://mathworld.wolfram.com/StereographicProjection.html
• http://www.sosmath.com/trig/Trig2/trig2/trig2.html
• http://www.sosmath.com/algebra/logs/log4/log4.html