Spring 2013, TuTh 1-2:30, Science Center 309
My name: Andrew Cotton-Clay (please call me Andy)
Office: Science Center 527
Office Hours: Wed 1:15-2:15 and Thu 2:45-3:45 or by appointment.
E-mail: acotton at math
Course Assistants:
Felix Wong, fwong at college
Anirudha Balasubramanian, balasubramanian at college
CA Sections and Office Hours:
Felix: Section Th 4-5, location TBA; Office hours W 9-11 PM in
Winthrop
D Hall
Anirudha: TBA
Syllabus: Math 113 Syllabus
Course iSite: Math 113 iSite (Look for Felix's lecture notes here!)
Due Feb 7
Due Feb 14
Due Feb 21
Due Feb 28
Due Mar 7
Midterm Practice Problems (solutions)
The midterm will cover chapters 1-9 of the textbook [BN], excluding
material we did not cover (e.g. only Morera's theorem from
chapter 7). The set of practice problems is representative of the sort of
problem that will appear on the midterm, but is longer by a problem or
two.
Due Mar 28
Due Apr 4
Due Apr 11 or 16
Due Apr 18 or 23
Due Apr 25 or 30
Tue 1/29: BN 1.1-2. Algebra and geometry of the complex numbers. Roots of
unity.
Thu 1/31: BN 1.3-4, 2.2-3. Stereographic projection exchanges circles and
circles & lines. Inversion and coordinates at infinity on the Riemann
sphere. Radius of convergence for power series and derivatives of power
series.
Tue 2/5: Uniqueness properties for power series (remainder of BN 2.3).
Cauchy-Riemann equations; proof that these plus continuous partials
implies holomorphic: see BN 3.1. CR equations equivalent to derivative
matrix commutes with rotation by pi/2. Holomorphic polynomials in x and y
are precisely those which can be written in terms of z and z-bar with no
z-bars: BN 2.1.
Thu 2/7: Exponential, trigonometric, and logarithm functions of a complex
variable: see BN 3.2 for the first two and BN 8.2 (excluding the theorem)
for the logarithm. Line integrals of complex functions and the fundamental
theorem of such: see BN 4.1.
Tue 2/12: BN 4.2 and 5.1: Closed curve theorem, Cauchy's integral theorem,
and proof that holomorphic functions on a disk are analytic with radius of
convergence at least the radius of the disk (BN 6.1-2).
Thu 2/14: BN 5.2 and beginning of 6.3. Example of integral from 0 to
infinity of (sin x)/x via closed curve theorem. Liouville theorem and
generalizations, fundamental theorem of algebra, uniqueness theorem for
holomorphic functions, and mean value theorem.
Tue 2/19: BN 6.3 and beginning of 7.1: Maximum and minimum modulus
theorems via mean value theorem and via power series. Open mapping
theorem. BN 1.4 and 8.1: Topological details for the general closed curve
theorem (different method than book).
Thu 2/21: More topological details for general closed curve theorem.
Definition of connected; proof it's the same as piecewise linearly
connected for open sets in C. Piecewise linear model for the first
homology of C in terms of formal sums of triangles, edges, and points. BN
7.2: Morera's theorem, limits of holomorphic functions, and Riemann Zeta
function is holomorphic for Re(z)>1.
Tue 2/26: Open sets on Riemann sphere are sets whose intersection with C
is open, plus, if the set includes infinity, it includes a neighborhood of
infinity: i.e. all z with |z|>R, for some R. Conclusion of theorem that
the closed curve theorem applies to open sets U in C such that C-hat minus
U is connected. BN 7.2: More applications of Morera's theorem: the Gamma
function is holomorphic for Re(z)>0, and removal of singularities. BN 9.1:
Removable singularities, poles, and essential singularities.
Casorati-Weierstrass theorem. BN 9.2: Laurent expansions: definition and
convergence.
Thu 2/28: BN 9.2: Laurent expansions and proof they exist for functions
defined in an annulus. Definition of meromorphic functions, and their
nature as holomorphic maps to the Riemann sphere; proof that holomorphic
maps from the Riemann sphere to itself are rational functions (not in the
book). BN 10.1: Definition of the residue of a function at a pole in terms
of its Laurent expansion and in terms of the integral of the function
around a small loop.