ROWAN UNIVERSITY
Department of Mathematics
Syllabus
Math 01.513 - Complex Analysis II
CATALOG DESCRIPTION:
Math 01.513 Complex Analysis II
3. s.h.
(Prerequisites: Math 01.512 Complex Analysis I)
This course will cover advanced topics in complex analysis:
Laurent series, meromorphic functions, conformal mappings, analytic continuation,
fractional linear transformations and elliptic functions.
CONTENT:
1. Laurent Series
1.1 Residue theorems
1.2 Application to
Real Analysis
1.3 Behavior of functions
in neighborhoods of isolated singularities
1.4 Casorati-Weierstrass
Theorem and Picard's Theorem
2. Entire Functions
2.1 Fundamental properties
2.2 Picard's First
Theorem
2.3 Infinite products
2.4 Weierstrass'
Factor Theorem
3. Meromorphic Functions
3.1 Poles and zeros
of Meromorphic Functions
3.2 Rational functions
3.3 Muttag-Leffler
Theorem
3.4 The Gamma Function
4. Conformal Mappings
4.1 Analyticity from
a mapping point of view
4.2 Elementary mapping
problems
4.3 Critical points
and magnification
4.4 Riemann Mapping
Theorem
5. Analytic Continuation
5.1 Uniqueness of
Analytic Continuation
5.2 Natural Boundary
5.3 Principle of
Reflection
5.4 Monodromy Theorem
6. Fractional Linear Transformations
6.1 Group properties
and matrix representations
6.2 Invariance, fixed
points and inversions
6.3 Cross ratios
6.4 F.L.T. of a half
plane into the interior of a circle
7. Periodic Functions
7.1 Simple periodic
functions
7.2 Doubly periodic
functions
7.3 Period points
7.4 Elliptic functions
8. Special Topics
8.1 Rouche's Theorem
8.2 Hurwitz's Theorem
8.3 Schwartz's Theorem
8.4 Riemann surfaces
and multiple-valued functions
TEXTS:
Boas, R.P., INVITATION TO COMPLEX ANALYSIS, Random House,
New York, 1987.
Churchill, Brown and Verhey, COMPLEX VARIABLES AND APPLICATIONS,
5th ed., McGraw-Hill, NY, 1990.
Rev:2-01/ HN
C:\Grd.revise.syl.complexanaII.grd