Math 4200-1
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Week 1: Aug 20-24
aug20.pdf 1.1 the complex plane
aug22.pdf 1.1-1.2 powers, roots, Euler, identities, estimates.
aug24.pdf 1.3 basic complex transformations
Week 2: Aug 27-31
aug27.pdf 1.4 part 1: sets and sequences.
aug29.pdf 1.4 part 2: functions
aug31.pdf 1.5 part 1: complex differentiability
Week 3: Sept 5-7
sept5.pdf 1.5 part 2: chain rules and the inverse function theorem.
sept7.pdf 1.5 part 3: conformal maps example
Week 4: Sept 10-14
sept10.pdf 1.5, 1.4: connectivity and application, harmonic conjugates
sept12.pdf 1.6: common analytic and entire functions, branches of their inverses and related composition function branches.
sept14.pdf 2.1: complex and contour integrals
Week 5: Sept 17-21
sept17.pdf 2.1-2.2 contour integrals
sept19.pdf 2.1-2.2 contour integrals and Green's Theorem
sept21.pdf 2.2 contour integrals and sufficient conditions to guarantee an antiderivative.
Week 6: Sept 24-28
sept24.pdf 2.3 Local antiderivative theorem, and Cauchy in a Rectangle.
sept26.pdf 2.3 Global antidifferentiation in simply connected domains, and the deformation theorem.
sept28.pdf 2.3-2.4 Deformation theorem example; winding number.
Week 7: October 1-5
oct1.pdf 2.3-2.4 an example of the deformation theorem, and the index theorem
oct5.pdf 2.4 Index theorem finished, and Cauchy integral formula.
Week 8: October 15-19
oct15.pdf 2.4 Applications of Cauchy integral formula: Liouville's Theorem and the fundamental theorem of algebra
oct17.pdf 2.4 Fundamental theorem of algebra and Morera's Theorem, and non-centered Cauchy estimates for nth derivatives.
oct19.pdf 2.5 Mean value property and the maximum modulus principle.
Week 9: October 22-26
oct22.pdf 2.5 Mean value property and the maximum modulus principle.
oct24.pdf 2.5 and Poisson Integral formula.
oct26.pdf 3.1 locally uniform limits of sequences of analytic functions are analytic, and derivatives converge as well.
Week 10: October 29 - November 2
oct29.pdf 3.2 Power series yield analytic functions inside the radius of convergence
oct31.pdf 3.2 Analytic functions have power series which converge inside any disk of analyticity.
nov2.pdf 3.2-3.3 Radius of convergence is determined by domain of analyticity; introduction to Laurent series.
Week 11: November 5 - November 9
nov5.pdf 3.3 Laurent Series
nov7.pdf 3.3-4.1 Laurent Series and computing residues
nov9.pdf 3.3, 4.2 Laurent Series and, at the very end, a proof of the Residue Theorem.
Week 12: November 12 - November 16
nov 12 (no notes) Professor Triebergs lectured on 4.2-4.3
nov14.pdf 4.2-4.3 more contour integration
nov16.pdf review sheet and practice exam.
Week 13: November 19-21
nov21.pdf 4.3-4.4 more contour integration
Week 14: November 26-30
nov26.pdf 4.4,5.1 contour integration and infinite sums; begin conformal transformations.
nov28.pdf 5.2 fractional linear transformations
nov30.pdf 5.2 more fractional linear transformations, and the Riemann Sphere as a complex manifold.
Week 15: December 3-7
dec3.pdf 5.2+ applications of FLT's to PDE and geometry
dec5.pdf magic formulas for the gamma function and infinite products