Class Diary for M443, Spring 2019, Jochen Denzler
Wed Jan 009:
Intro to complex numbers: motivation, field axioms,
construction. See Notes. Hwk: The four hwk
problems imbedded in the notes are due Monday.
Fri Jan 11:
complex conjugate, polar representation of cplx numbers, geometric
interpretation. Hwk: remaining problems from handed out
list due Wed.
Mon Jan 14:
topological notions: limit point, neighborhood, limit; open and closed sets;
Bolzano-Weierstrass.
Wed Jan 16:
Riemann sphere and the point at infinity. Examples of simple functions and
their mapping properties.
Hwk: Pg 12, #25 due Wed; tody's due extended to Fri
Fri Jan 18:
open, closed, connected, simply connected sets; curves; Jordan curves;
continuous functions; definition of differentiable function.
Mon Jan 21:
MLK DAY
Wed Jan 23:
Complex differentiable functions: Examples; comparison with real
differentiability; Cauchy-Riemann Differential Equations;
Hwk: book page 20ff, numbers 1,12, 19,21 due Monday.
Hwk problems from sheet on mappings (number 7-10)
due Wednesday.
Fri Jan 25:
Holomorphic functions; preview without proof of key properties (in constrast to
differentiable real functions; conformal mapping by holomorphic
function (proof to come Monday)
Mon Jan 28:
Conformal mappings (proof and explanation). Linear mappings C-> C; Möbius
transforms. Integration begun (preview/summary).
Hwk from 3rd sheet due next Monday.
Wed Jan 30:
Definition of complex curve integral; basic properties and some examples;
fund'l theorem of calculus applies if integrand is a derivative. Dependence on
path or not?
Fri Feb 01:
Examples: no antiderivatve of 1/z on C setminus {0}. No antiderivative for Re
z because integral does depend on path. -- Comments on homework (mappings)
Mon Feb 04:
integrals over holomorphic functions are path independent; proof.
Wed Feb 06:
Comparison with Green's/Stokes' theorem. Started Cauchy integral formula.
Fri Feb 08:
Cauchy integral formula proved; consequences: infinite differentiability of
holomorphic functions (CIF for n-th derivative); path independent integral
implies holomorphic. Homework from book page 73-74: numbers 10, 16, 18 due
next Friday.
Mon Feb 11:
Consequences of Cauchy Integral Formula: Liouville's theorem. Fundamental
Theorem of Algebra.
Wed Feb 13:
comments on hwk. Overview: Def. of analytic functions in terms of power
series. Spoiler alert: holomorphic <=> analytic. Comparison with Calc 2 on real
series. (All details to be filled in yet.)
Fri Feb 15:
Series; absolute convergence. Rearrangement of abs. conv. series; Cauchy
product; geometric series. Power series
Mon Feb 18:
Uniform convergence vs pointwise convergence of series
Wed Feb 20:
Weierstrass Thm about termwise differentiation of uniformly (on closed bounded
sets) convergent series
Fri Feb 22:
Weierstrass Thm about termwise differentiation of uniformly (on closed bounded
sets) convergent series
Mon Feb 25:
Power series; converge in discs: radius of convergence; exponential, sine, and
cosine series. Hwk: pblms 15-18 due next
Monday.
Wed Feb 27:
Hint for pending hwk. Properties of exp, sin, cos obtained as consequences from
the defining series;
Fri Mar 01:
trig and hyp finished up; radius of convergence by Cauchy Hadamard; proof
started.
Mon Mar 04:
proof of Cauchy-Hadamard finished. Examples of functions defined by power
series: ln (1+x) = x-x^2/2+x^3/3-+... defines a holomorphic `logarithm'
function in a unit disc centered at 1. Bessel functions J_k.
Taylor series of a holomorphic function started.
Wed Mar 06:
EXAM 1
Fri Mar 08:
Discussion of exam
Mon Mar 11:
holomorphic functions are rep'd by their Taylor series. (proof and details);
comparison with theory in real variables from Calc 2
Wed Mar 13:
Singularity on boundary of disc of convergence; Abel's theorem in case of
convergence on the boundary (stated w/o proof). Identity theorems for analytic
functions. Hwk: pg 104 in book, numbers 3,4,6,7,10 due Wednesday after
spring break. Pg 120 numbers 6,7,8ace, 9 due Friday after spring
break.
Fri Mar 15:
Zeros of holo fcts are isolated; order of zero. Identity theorems proved. Open
mapping thm stated.
Mon Mar 18:
SPRING BREAK
Wed Mar 20:
SPRING BREAK
Fri Mar 22:
SPRING BREAK
Mon Mar 25:
Root functions started. Theorem on inverse functions stated without proof.
Refer to my own notes more than Ch 9 of text book.
Wed Mar 27:
root functions constructed in half planes
-- EVENING 5:15 option for a second attempt at (a new) exam 1. ROOM 123
Fri Mar 29:
a bit on exam 1; root functions
Mon Apr 01:
logarithm functions in simply connected domains; general power function
Wed Apr 03:
inverse function constructed by formal power series
Fri Apr 05:
Convergence proof for inverse function power series
Mon Apr 08:
holom and injective implies f'(z) not 0; Open mapping theorem; Laurent series
started.
Wed Apr 10:
Holomorphic functions in an annulus can be expanded as Laurent series.
Classification of isolated singularities.
Fri Apr 12:
Theorems about the 3 kinds of isolated
singularities. Homework due Wed
Mon Apr 15:
Wed Apr 17:
Fri Apr 19:
GOOD FRIDAY
Mon Apr 22:
EXAM 2
Wed Apr 24:
Fri Apr 26:
Mon Apr 29: STUDY DAY
Fri May 03: FINAL EXAM 12:30-2:30pm
(scheduled by university policy)
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