Revised August 2002, T. Luo, M. Vancliff

Course: Math 5322  Complex Variables

Complex numbers, analytic functions, contour integration, power series,

conformal mappings of simply-connected regions, analytic continuation.

Comprehensive Examination Material: part (b) on Math 5322 (Complex Variables)

1. Complex Numbers

(a) arithmetic and algebra of C, conjugation, absolute value

(b) Argand diagram, exponential form

(c) Riemann sphere

2. Complex Functions

(a) limits, continuity

(b) analytic (holomorphic) functions, Cauchy-Riemann equations

(c) polynomial and rational functions

(d) Fundamental Theorem of Algebra

(e) Power series, Laurent series, Abel's Limit Theorem

(f) exponential, logarithmic and trigonometric functions

3. Mappings

(a) connectedness, compactness

(b) Bolzano-Weierstrass Theorem

(c) conformal mappings, harmonic functions

(d) linear transformations

(e) Möbius Transformations

4. Contour Integration

(a) line integrals

(b) Cauchy's Theorem

(c) Cauchy's integral formula, winding numbers

(d) Liouville's Theorem

(e) Theorems of Morera and Goursat

(f) removable singularities and Taylor's Theorem

(g) classification of isolated singularities

(h) zeros and poles, meromorphic functions

(i) the argument principle

(j) maximum (modulus) principle

(k) Scharz' Lemma, Theorems of Hadamard and Phragmén-Lindelöf

(l) calculus of residues, residue formula

(m) harmonic functions

(n) mean-value property

(o) Riemann-Mapping Theorem

(p) conformal mappings of simply-connected regions

5. Analytic Continuation

(a) germs

(b) Riemann surfaces, branch points

Books:

1. J. B. Conway, Functions of One Complex Variable, 2nd Ed., Graduate

Texts in Mathematics, Vol 11, Springer Verlag, August 1978.

2. L. V. Ahlfors, Complex Analysis, 3rd Ed, Mc-Graw Hill College Div,

January 1979.

3. I. Stewart and D. O. Tall, Complex Analysis, Cambridge Univ Press,

May 1983.

4. E. C. Titchmarsh, The Theory of Functions, Oxford University Press,

2nd Ed, December 1939.