MAT 452 -- Complex Analysis  
Course Description

Hadamard has written that "the shortest route between two truths in the real domain passes through the complex plane." This course will trace the development of the powerful techniques which complex analysis can bring to bear on a wide variety of mathematical problems. In support of Hadamard's famous statement, one finds an astonishing diversity of applications, ranging from computer science (analysis of algorithms), to enumerative combinatorics (growth rate of sequences), to engineering (the Joukowski transformation and airfoil design; electrical potentials, heat conduction, and hydrodynamics), to physics (differential equations, Fourier and Laplace tranforms) and number theory (Mellin transforms and exponential sums).

Text: H. A. Priestley, "Introduction to Complex Analysis," (revised ed.) Oxford University Press, New York, 1990.

References:

• Joseph Bak and Donald J. Newman, "Complex Analysis," (2nd ed.) Springer-Verlag, New York, 1997.
• George Polya and Gordon Latta, "Complex Variables," John Wiley & Sons, New York, 1974.
• John H. Mathews and Russell W. Howell, "Complex Analysis for Mathematics and Engineering," Jones and Bartlett, Boston, 2001.
• Ruel V. Churchill and James Ward Brown, "Complex Variables and Applications," McGraw-Hill, New York, 1984.
• Jerrold E. Marsden & Michael J. Hoffman, "Basic Complex Analysis," (2nd ed.) W. H. Freeman and Company, New York, 1987.

Syllabus: We'll cover as much of Priestley as time permits, with applications interspersed throughout. In slightly more detail:

• The Complex Plane
• Holomorphic Functions and Power Series
• Cauchy's Theorem
• Singularities and Multifunctions
• The Maximum Modulus Theorem
• Residue Theory
• Evaluating Sums and Integrals via Residues
• Laplace and Fourier Transforms
• Conformal Maps and Harmonic Functions