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Complex Analysis
Math 4200 home page
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Lecture notes will be posted on this page, by noon the day before class. I recommend looking at the notes before class and bringing a copy with you, so we can refer to them together as we carefully fill in the details.
Week 1: Aug 22-26
aug22.pdf 1.1 the complex plane
http://en.wikipedia.org/wiki/Complex_analysis has a nice overview of complex analysis, with links.
aug24.pdf 1.2 properties and estimates for complex number operations.
aug26.pdf 1.3 Basic complex functions.
Week 2: Aug 29 - Sept 2
aug29.pdf 1.4 part 1: sets and sequences.
aug31.pdf 1.4 part 2: functions.
sept2.pdf 1.5 complex analytic functions
Week 3: Sept 7 - Sept 9
sept7.pdf 1.5 continued: the chain rule and inverse function theorem.
sept9.pdf 1.5 continued: the chain rules and the rotation-dilation differential maps for analytic functions.
Week 4: Sept 12 - Sept 16
sept12.pdf 1.5 harmonic functions and harmonic conjugates; connectivity and path connectivity from 1.4
http://en.wikipedia.org/wiki/Harmonic_function
sept14.pdf 1.6 differentiation of basic functions; branches and branched domains for multivalued functions revisited.
sept16.pdf 2.1 contour integrals
Week 5: Sept 19 - Sept 23
sept19.pdf 2.1-2.2 contour integrals, estimates, and the implications of Green's Theorem.
sept21.pdf 2.2 contour integrals, path independence, and antiderivatives in simply-connected domains.
sept23.pdf 2.3 Cauchy's antiderivative theorem improved.
Week 6: Sept 26 - Sept 30
sept26.pdf 2.3 continued
sept28.pdf 2.3 continued, logarithms on simply connected domains.
sept30.pdf 2.4 winding numbers for curves and Cauchy's Integral Formula.
Week 7: Oct 3 - 7
oct3.pdf review sheet.
oldexam1.pdf from 2007
ourexam.pdf
ourexamsolutions.pdf
oct7.pdf 2.4: applications of the Cauchy Integral formula
Week 8: Oct 17 - 21
oct17.pdf 2.4-2.5 continued
http://en.wikipedia.org/wiki/Riemann_zeta_function one of the more magical complex analytic functions
oct19.pdf 2.5 continued - maximum modulus principles for analytic functions; max/min principles for harmonic functions.
Poisson.pdf Poisson.mw harmonic functions on the unit disk, via the Poisson kernel.
oct21.pdf 2.5 continued - applications of the maximum modulus principle, and proof of the Poisson Integral formula using Mobius transformations....Mobius transformations are the isometries of the hyperbolic disc:
http://en.wikipedia.org/wiki/Hyperbolic_geometry
http://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane
Week 9: Oct 24 - 28
oct24.pdf review; then begin 3.1: sequences and series of analytic functions.
oct26.pdf 3.1-3.2 convergence and term by term differentiation of power series; Taylor series.
oct28.pdf 3.2 convergence of Taylor's series, and consequences.
Week 10: Oct 31 - Nov 4
oct31.pdf 3.2-3.3; multiplying Taylor series; introduction to Laurent Series.
http://www.math.utah.edu/ugrad/colloquia.html Wednesday's undergraduate colloquium about the Riemann Zeta function
nov2.pdf 3.3 existence and uniqueness of Laurant series for functions analytic in annuli; residues.
nov4.pdf 3.3 characterizations of isolated singularities.
Week 11: Nov 7-11
nov7.pdf 4.1-4.2 the residue theorem, and methods for computing residues.
nov9.pdf 4.1-4.2 residues; the residue theorem for exterior domains, and general change of variables in contour integrals.
nov11.pdf 4.3 real variable definite integrals via the residue theorem magic.
Week 12: Nov 14-18
nov14.pdf review notes (with lots of blank space) for the exam on Wednesday. See also
oldexam2.pdf from 2007
exam2.pdf our exam
exam2sols.pdf solutions
nov18.pdf 4.3-4.4: contour integrals for PV integrals; magic formulas for series using pi*cot(pi*z).
Week 13: Nov 21-23
nov21.pdf 4.4 series formulas; infinite partial fraction expansions for meromorphic functions.
nov23.pdf 8.1-8.2 revisiting Laplace transform from 2280/2250 - with a residue formula for the inverse Laplace transform.
Week 14: Nov 28 - Dec 2
nov28.pdf 5.1-5.2 conformal transformations.
nov30.pdf 5.2 fractional linear transformations
dec2.pdf 5.2 continued, and the Riemann sphere as a complex manifold.
Riemann surfaces
FLT's (Mobius transformations)
Youtube video! How the mysterious actions of Mobius transformations on the complex plane really arise from rotations and translations of the unit sphere, combined with stereographic projection
Week 15: Dec 5 - Dec 9
Monday Dec 5: Mi Ryu, Geoff, and Mi Jeong gave an overview of how to prove the prime number theorem with the tools of complex analysis and the Riemann Zeta function. Their presentation was based on notes they found, by Terence Tao at UCLA. As it turns out, Tao is a well known and highly respected mathematicican. Here's his home page, with links some of you might find interesting and/or helpful: http://www.math.ucla.edu/~tao
Wednesday Dec 7: Mike constructed the Weierstrasse P functions and showed how they parameterize complex tori satisfying the implicit cubic equation w2=4 z3-a z - b in C2.
Friday Dec 9: Leah, Melissa and Trevor will tell us interesting things about Julia sets and the Mandelbrot set.
finalreview.pdf review notes for our course.
2002finalexam.pdf Final exam from 2002