| Week |
Date of Monday |
Topics |
Lecture notes |
Example sheet |
Comments |
| 1 |
Oct. 6 |
Administration. Overview of content. Notation and terminology.
Arguments and branches. Domains, examples. Domains are
path-connected. Contours, closed, simple. Contour integrals,
the main example. Linearity in the integrand and in the domain
of integration. Reparametrisation. Elements and how to
remember them. Element of arc-length, the ML inequality.
Element of, and variation of, argument. Elements of radius, of
logarithm of radius, and of complex logarithm.
|
Tuesday
Wednesday
Friday
|
One
|
Tuesday: The lecture capture was not correctly set.
Will try to fix in future.
Wednesday: A student asks "What do the diamonds, in
the lecture notes, mean?" They signal the end of a definition,
example, or remark.
Friday: A student asks "What is an element?" "An
element of P" means "a little bit of P", whatever P is. We
feed them contours (which they differentiate) and integrate
the result.
|
| 2 |
Oct. 13 |
Winding numbers. Holomorphicity. (Real) total derivative of
holomorphic function. Holomorphic functions are conformal (away
from zeros of derivative). Sums, products, ratios, compositions
of holomorphic functions. Non-holomorphic functions. Sketching
exercises. Holomorphic implies continuous. Continuity of the
"error". Review of power series. Complex exponential and trig
functions. Their properties. Complex logarithm, has continuous
branches. Segments. Contour integrals of polynomials along
segments. Convexity, convex hulls, examples. Triangles,
boundaries of triangles, degeneracy, handedness (positive versus
negative). Integrals of polynomials about triangles, of
analytic functions. The midpoint subdivision, arc-length and
diameter of the parts.
|
Tuesday
Wednesday
Friday
|
Two
|
All week: Obviously the lecture capture is giving me
trouble. I will ask one student to volunteer each lecture to
make sure that all boards get captured.
|
| 3 |
Oct. 20 |
Linear functions (polynomials, analytic functions) integrate to
zero about triangles. Goursat's lemma: holomorphic functions
integrate to zero about triangles, about rectangles, about
polygons, about annuli (via reparametrisation by complex
exponential). Primitives. If a function integrates to zero
about triangles in a disk, then it has a primitive in the disk.
Counterexample to obvious generalisation. Singular
\(n\)-simplices, group of \(n\)-chains. Faces of simplices, the
boundary operator. The "square" of the boundary operator is
zero. Cycles, boundaries, homology classes.
|
Tuesday
Wednesday
Friday
|
Three
|
Wednesday: A student asks "Do holomorphic functions
integrate to zero about round disks?" Yes! Here is one proof:
take a sequence of annuli with inner radius tending to zero.
Now use the ML-inequality.
|
| 4 |
Oct. 27 |
Review of notation. Coning operator, convex sets have trivial
first homology. Midpoint subdivision operator (in dimensions
one and two). Boundary commutes with subdivision. Boundary
commutes with straightening. The homological version of
Cauchy's theorem. Proof of Cauchy's theorem. Review of
statements so far. Topological versus homological boundary.
Regions, triangulations of regions. Holomorphic functions
integrate to zero about the boundaries of regions.
|
Tuesday
Wednesday
Friday
|
Four
|
Wednesday: A student says "You made a mistake on
Tuesday when defining the coning operator." Oops! This is now
fixed (and written in red) in the handwritten lecture notes. It
is still wrong in the lecture capture.
|
| 5 |
Nov. 3 |
The reproduction formula. The mean value theorem. Analytic
functions. Series expansions. Analyticity implies
holomorphicity. The Cauchy estimate. Holomorphicity implies
analyticity. Series expansion coefficents are well-defined,
agree with derivatives, lower bound for radius of convergence.
The fundamental theorem of complex analysis (following Bers).
Zeros. Vanishes to infinite order implies vanishing
identically. Orders of zeros. Patching lemma. Factoring
lemma. Accumulation points, isolated sets. The set of zeros of
a holomorphic function is isolated in the domain of the
function.
|
Tuesday
Wednesday
Friday
|
Five
|
Wednesday: A typo in lecture: I should have written
\(\frac{1}{R_0} \geq \limsup_{n \to \infty} \sqrt[n]{|a_n|}\).
(That is, "greater than or equal", not "less than or equal".) I
have fixed this in the handwritten lecture notes. It is still
wrong in the lecture capture.
Wednesday: Various questions about homotopy and
deformation of contours.
|
| 6 |
Nov. 10 |
The identity theorem. Extensions and restrictions. Removing
removable singularities. Laurent series, inner and outer radii
of convergence. Laurent's theorem. Poles, simple poles, orders
of poles. Zeros, simple zeros, orders of zeros. Essential
singularities.
|
Tuesday
Friday
|
Six
|
Tuesday: The formula I wrote on the board for the inner
radius of convergence of a Laurent series was wrong. I fixed
this on Friday.
Wednesday: Power outage meant no lecture capture, so I
cancelled lecture. Instead we had a Q & A session.
|
| 7 |
Nov. 17 |
Trichotomy. Casorati-Weierstrass. Meromorphic functions,
patching, pactoring. Residue theorem. Improper definite
integrals, examples, the six stages of integration: convergence,
simplification, choose function and contour, compute residues,
estimate away "minor arcs", cross check. Homeomorphisms,
invariance of homology. Branch cuts.
|
Tuesday
Wednesday
Friday
|
Seven
|
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| 8 |
Nov. 24 |
Domains with trivial homology, primitives, branches of the
logarithm. A logarithmic integral. Entire functions.
Liouville's theorem. The fundamental theorem of algebra. The
maximum modulus principle. The minimum modulus principle. The
open mapping theorem.
|
Tuesday
Wednesday
Friday
|
|
Tuesday: In the proof of the Branches Theorem I used
very poor notation (capital C versus lower case c???) for the
constant of integration and its logarithm. This led to
confusion. I have changed this (to K versus k) in the
handwritten notes; I also colour-coded them.
Friday: There are many versions of the maximum and
minimum modulus principles. I gave slightly more complicated
versions in lecture; I give slightly simpler versions in the
typed lecture notes.
|
| 9 |
Dec. 1 |
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| 10 |
Dec. 8 |
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