Evangelos A. Coutsias, Math 316

Math 313: Complex Variables

Instructor: Evangelos A. Coutsias

Required Text: E.B. Saff and A.D. Snider Fundamentals of Complex Variables with applications to Engineering and Science
3rd Edition (ISBN 0-13-907874-6)

Supplement (recommended): Schaum's Outlines: Complex Variables Murray R. Spiegel (good source of problems and examples)

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Solutions to the Homework and Exams and select Lecture Notes

practice Exam I

(caution: this exam is based on a previous text and syllabus!)

Final Exam: xxx., xxx. xx, xx:00->xx:00

Outline

This course is good preparation for courses in advanced engineering mathematics and physics, quantum physics, electromagnetics and wave propagation, fluid flow and elasticity. Potential problems (electrostatics, linear elasticity, laminar flow) are solved in a very natural and straightforward way using techniques from complex variables such as conformal mapping. Difficult calculations in Classical Mechanics as well as inversions of Fourier and Laplace Transforms are carried out by the methods of Contour Integration. Several other applications will be presented. Multivalued functions, such as logarithms and roots, can only be understood properly in the context of Complex Analysis, and Taylor's theorem becomes much easier to state and comprehend.

Topics to be covered include:

Complex Algebra; Polar form of complex numbers; Euler's formula. Cauchy-Riemann equations and differentiability; Analytic functions; Exponentials, Logarithms; Triginometric functions and their inverses. Contour Integration; Cauchy's Theorem and Integral Formula. Power series and convergence; Taylor and Laurent series; Residue Theory; Zeros and Poles. Evaluation of Integrals by Residue Theory; Multivalued Functions, Branch Points and Branch Cuts; Rouche's Theorem and finding the zeros of a function; Inversion of Laplace Transforms. Elementary Conformal Mappings and Applications; Poisson Formula; Dirichler and Neumann problem on the disk and half-plane; Potential Theory.