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Math 4200-001
4200-1 home page
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It is my goal to have lecture note outlines for each day posted by noon the day before. In case you have to miss class I will also post filled in versions after class. On CANVAS you can access recordings of our lecture meetings.
Week 1: August 24-28 Sections 1.1-1.3
aug24 1.1-1.2 pre.pdf 1.1-1.2 introduction to algebra and geometry in the complex plane. aug24 1.1-1.2 post.pdf after-class version.
aug26 1.1-1.2 pre.pdf 1.1-1.2 solutions to polynomial equations aug26 1.1-1.2 post.pdf after-class version.
aug28 1.2-1.3 pre.pdf 1.2-1.3 intro to transformations of the complex plane aug28 1.2-1.3 post.pdf after-class version.
Week 2: August 31 - September 4 Sections 1.3-1.5
aug31 1.3 pre.pdf 1.3 basic complex transformations aug31 1.3 post.pdf after-class version.
sept2 1.5 pre.pdf 1.5 complex differentiability sept2 1.5 post.pdf after-class version.
sept4 1.5 pre.pdf 1.5 chain rules and rotation-dilation differential map sept4 1.5 post.pdf after-class version.
Week 3: September 9-11 Section 1.5
sept9 1.5 pre.pdf 1.5 3220 apps: differential map, CR theorem, inverse function theorem. sept9 1.5 post.pdf after-class version.
sept11 1.5-1.6 pre.pdf 1.5 harmonic functions and harmonic conjugates. sept11 1.5 post.pdf after-class version.
Week 4: September 14-18 Sections 1.5-2.1
sept14 1.5-1.6 pre.pdf 1.5-1.6: harmonic conjugates; basic analytic functions. sept14 1.5-1.6 post.pdf after-class version.
sept16 1.6 pre.pdf 1.6: compositions and inverses of analytic functions; fundamental domains. sept16 1.6 post.pdf after-class version.
sept18 1.6-2.1 pre.pdf 2.1 integrals over curves for complex-valued functions. sept18 1.6-2.1 post.pdf after-class version.
Week 5: September 21-25 Sections 2.1-2.2
sept21 2.1-2.2 pre.pdf contour integrals and the FTC for analytic integrands. sept21 2.1-2.2 post.pdf after-class version.
sept23 2.2 pre.pdf Cauchy's Theorem and the Replacement Theorem for contour integrals of analytic functions. sept23 2.2 post.pdf after-class version.
sept25 2.2 pre.pdf Path independence and antiderivatives sept25 2.2 post.pdf after-class version.
Week 6: September 28 - October 2 Sections 2.2-2.3
sept28 2.2-2.3 pre.pdf antiderivatives for C1 analytic functions in simply-connected domains sept28 2.2-2.3 post.pdf after-class version.
sept30 2.3 pre.pdf rectangle lemma and local antiderivatives for analytic functions. sept30 2.3 post.pdf after-class version.
oct2 2.3 pre.pdf homotopies, simply-connected, antiderivatives, deformation theorems oct2 2.3 post.pdf after-class version.
Week 7: October 5-9 Section 2.4
oct5 2.4 pre.pdf winding number and Cauchy Integral Formula oct5 2.4 post.pdf after-class version.
oct7 2.4 pre.pdf Cauchy integral formula for derivatives; exam review. oct7 2.4 post.pdf after-class version.
Week 8: October 12-16 Section 2.4
oct12 2.4 pre.pdf C.I.F. for derivatives, Liouville's Theorem, Fundamental Theorem of Algebra. oct12 2.4 post.pdf after-class version.
oct14 2.4 pre.pdf Fundamental Theorem of Algebra, Morera's Theorem and consequences, Zeta function. oct14 2.4 post.pdf after-class version.
oct16 2.5 pre.pdf mean value property and maximum principle for analytic and harmonic functions. oct16 2.5 post.pdf after-class version.
Week 9: October 19-23 Sections 2.5-3.1
oct19 2.5 pre.pdf maximum principles for analytic and harmonic functions; conformal diffeomorphisms of the unit disk. oct19 2.5 post.pdf after-class version.
oct21 2.5 pre.pdf Poisson integral formula for harmonic functions, via Mobius transformations. oct21 2.5 post.pdf after-class version.
oct23 3.1 pre.pdf sequences and series of analytic functions oct23 3.1 post.pdf after-class version.
Week 10: October 26-30 Sections 3.2-3.3
oct26 3.2 pre.pdf existence, uniqueness and convergence of power series for analytic functions. oct26 3.2 post.pdf after-class version.
oct28 3.2 pre.pdf Taylor series tricks oct28 3.2 post.pdf after-class version.
oct30 3.3 pre.pdf Laurent series. oct30 3.3 post.pdf after-class version.
Week 11: November 2-6 Sections 3.3-4.2
nov2 3.3 pre.pdf Laurent series and classification of isolated singularities nov2 3.3 post.pdf after-class version.
nov4 3.3 pre.pdf Laurent series and classification of isolated singularities nov4 3.3 post.pdf after-class version.
nov6 4.1-4.2 pre.pdf Residue theorem and residue computation shortcuts nov6 4.1-4.2 post.pdf after-class version.
Week 12: November 9-13 Sections 4.2-4.3
nov9 4.2 pre.pdf Residue Theorems and exam review notes nov9 4.2 post.pdf after-class version.
nov11 hwandreview pre.pdf homework and review questions nov11 hwandreview post.pdf after-class version.
Week 13: November 16-20 4.3-4.4
nov16 4.3 pre.pdf integral applications of contour integration nov16 4.3 post.pdf after-class version.
nov18 4.3 pre.pdf continued nov18 4.3 post.pdf after-class version.
nov20 4.4 pre.pdf magic formulas for summing series. nov20 4.4 post.pdf after-class version.
Week 14: November 23-25 5.1-5.2
nov23 5.1-5.2 pre.pdf conformal transformations between domains nov23 5.1-5.2 post.pdf after-class version.
nov25 5.1-5.2 pre.pdf conformal transformations continued. nov25 5.1-5.2 post.pdf after-class version.
Week 15: November 30 - December 4 5.2 and presentations
nov30 5.2 pre.pdf FLT examples; Riemann sphere revisited as a Riemann surface. nov30 5.2 post.pdf after-class version. Also, Daniel presented about the meromorphic functions on complex tori.
December 2: Jess, Carlie and Austin presented on the Prime Number Theorem and the Riemann Zeta function.
December 4: finalereview.pdf Also, Aidan and Emma presented on the general Schwarz reflection principle, and on other analytic continuation ideas related to the Zeta function. finalereview post.pdf