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
(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,
January 1979.
3.
May 1983.
4. E. C. Titchmarsh,
The Theory of Functions,
2nd Ed, December 1939.