Class Diary for M443, Spring 2013, Jochen Denzler
Wed Jan 09:
Introduction; construction of complex numbers and field
axioms. Hwk: as handed out due Monday,
tentatively
Fri Jan 11:
polar representation of complex numbers; cis; geometric interpretation;
Hwk pg12, #25
Mon Jan 14:
topological notions: neighborhood, limit point, limit, Bolzano-Weierstrass;
Cauchy criterion. Hwk pg20ff # 1,12,19,21
Wed Jan 16:
Riemann sphere; the point infinity; open sets; connected sets; domains;
Examples of simple functions and their mapping properties: linear and z^2
Fri Jan 18:
simply connected domains; continuous functions; differentible functions defined
and two examples. Hwk from handed out notes
on mapping properties due next Fri
Mon Jan 21:
MLK DAY
Wed Jan 23:
no such thing as `multivalued function': explanation about usage of this term
in the book. -- Cauchy-Riemann differential equations.
Fri Jan 25:
ice rain cancellation
Mon Jan 28:
Cauchy-Riemann DE's part 2, and conformality
Hwk from handed-out notes due Fri or Mon
Wed Jan 30:
Hints for Hwk 9; integrals of (continuous) complex functions along a curve (of
finite length; say piecewise differentiable); example.
Fri Feb 01:
path independence theorem stated, idea outlined, details begun: polygonal
curves and approximation by them.
Mon Feb 04:
How to practically decompose a polygonal path into triangles; proof that this
can always be done.
Wed Feb 06:
Path independence theorem proved.
Fri Feb 08:
Continuous deformation of integration paths leave integral of holomorphic
functions unchanged. Hwk: pg.73ff #10,16,18
Mon Feb 11:
Cauchy's integral formula
Wed Feb 13:
holomorphic => infinitely many derivatives
Fri Feb 15:
Liouville's thm (in book, that's only on pg 141). Fundamental thm of algebra
Mon Feb 18:
Harmonic functions
Wed Feb 20:
Series of complex numbers; convergence and absolute convergence
Fri Feb 22:
series; adding and multiplying.
Mon Feb 25:
EXAM 1
Wed Feb 27:
series of functions; uniform convergence
Fri Mar 01:
uniform convergence cont'd
Mon Mar 04:
uniform convergence (on compact sets) of differentiated series
Wed Mar 06:
differentiated series finished; power series Hwk:
Set 4
Fri Mar 08:
radius of convergence; implications for adding / multiplying power series
Mon Mar 11:
Cauchy-Hadamard. Hwk pg 104f #3,4,6,7,10 Abel thm (pg 105 #11) stated
without proof.
Wed Mar 13:
exponential and trig and hyperbolic functions. Hwk: pg 120, #6,7,8ace,9
Fri Mar 15:
Taylor series; holomorphic functions are representable as power series
Mon Mar 18:
Singular point on boundary of disk of convergence. Uniqueness theorems.
Wed Mar 20:
EXAM 1 (second chance)
Fri Mar 22:
SPRING BREAK
Mon Mar 25:
SPRING BREAK
Wed Mar 27:
SPRING BREAK
Fri Mar 29:
GOOD FRIDAY
Mon Apr 01:
Practical calculation with power series: long division; composition.
Hwk set 5 due Friday or Monday (you'll
benefit in class from doing it asap)
Wed Apr 03:
injectivity. The problem with roots in C. Domains of root fcts
Fri Apr 05:
Root fcts constructed in various domains based on real variables formulas;
logarithm likewise. Which domains support holomorphic root and logarithm
functions. Power series for ln(1+z) and binomial series.
Mon Apr 08:
Inverse functions; general theorem explained, and proof via power series
begun.
Wed Apr 10:
Convergence proof for inverse function series
Fri Apr 12:
correction; Hwk: read section on Riemann surfaces in book pg
133/134. Laurent series started
Mon Apr 15:
Representation of holomorphic functions in annuli by Laurent series;
isolated singularities, and the purpose of studying them
Wed Apr 17:
Removable singularities; poles; bounded isolated singularities are removable
Fri Apr 19:
EXAM 2
Mon Apr 22:
intro to new hwk;
essential singularities (Casorati-Weierstrass); residues
Wed Apr 24:
Fri Apr 26:
Mon Apr 29: STUDY DAY
Thu May 02: FINAL EXAM 12:30-02:30
(scheduled by university policy)
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