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Math 617
Fall 2018 Journal

Wednesday, December 12
The final exam took place, and solutions are available.
Tuesday, December 4
We reviewed for the final exam and previewed some topics from Math 618. The slides from class are available.
Thursday, November 29
We discussed the maximum-modulus theorem and the Schwarz lemma. The slides from class are available.
Tuesday, November 27
We worked in groups on the computation of integrals via the residue theorem. The slides from class are available.
Tuesday, November 20
We discussed the notion of residue, the residue theorem, and a sample application to the evaluation of definite integrals. The slides from class are available.
Thursday, November 15
We discussed the Casorati–Weierstrass theorem and the existence of Laurent series of functions in annuli. We worked in groups on Exercise 1 in Section 1 of Chapter V. The slides from class are available.
Tuesday, November 13
We discussed the classification of isolated singularities, Riemann’s removable singularities theorem, the order of poles, and Picard’s great theorem about essential singularities. The assignment is shown on the last page of the slides from class.
Thursday, November 8
The second exam was given, and solutions are available.
Tuesday, November 6
We reviewed for the exam to be given next class and discussed the local behavior of analytic functions as mappings. The slides from class are available.
Thursday, November 1
We discussed the symmetric form of Rouché’s theorem and the local injectivity of analytic functions with nonzero derivative. The assignment is shown on the last page of the slides from class.
Tuesday, October 30
We discussed the argument principle and Rouché’s theorem. The assignment is shown on the last page of the slides from class.
Thursday, October 25
We discussed simple connectivity of regions and logarithms of functions. The assignment is shown on the last page of the slides from class.
Tuesday, October 23
We looked at Morera’s proof of his theorem, the concept of homotopy, and the ideas in the proof of the path-deformation principle. The assignment is shown on the last page of the slides from class.
Thursday, October 18
We worked through Dixon’s 1971 proof of the homology version of Cauchy’s integral formula, worked an example, and saw the statement of Morera’s converse of Cauchy’s theorem. The assignment is shown on the last page of the slides from class.
Tuesday, October 16
We discussed the concept of winding number and the statement of the homology version of Cauchy’s integral formula. The assignment is shown on the last page of the slides from class.
Thursday, October 11
We worked together on proving that the zeros of a nonconstant analytic function are countable, saw how identities on the real line propagate to the complex plane, and looked at some pictures as motivation for the upcoming topic of winding numbers. The assignment is shown on the last page of the slides from class. Thanks to Paul for pointing out that “functions” should say “analytic functions” in the problem on the qualifying examination.
Tuesday, October 9
We discussed the existence of power series expansions for analytic functions, the example of Bernoulli numbers, and the property of zeros of analytic functions being isolated. The assignment is shown on the last page of the slides from class.
Thursday, October 4
The first exam was given, and solutions are available.
Tuesday, October 2
We reviewed for the exam to be given next class, and we proved the fundamental theorem of algebra and Liouville’s theorem by applying Cauchy’s integral formula on disks. The slides from class are available.
Thursday, September 27
We derived various properties of analytic functions on disks: namely, the mean-value property, Cauchy’s integral theorem, and Cauchy’s integral formula. The tools were the definition of a path integral, the Cauchy–Riemann equations, and the Möbius transformations that map the unit disk onto itself. The assignment (not to hand in) is shown on the last page of the slides from class.
Tuesday, September 25
We discussed the notion of conformality. The assignment is shown on the last page of the slides from class. Reminder: The first exam takes place on Thursday, October 4.
Thursday, September 20
We discussed some terminology for paths/curves: smooth, piecewise smooth, rectifiable. The assignment is shown on the last page of the slides from class.
Tuesday, September 18
We discussed the complex exponential, sine, and cosine functions; their relationships; and the notion of a branch of the logarithm. The assignment is shown on the last page of the slides from class.
Thursday, September 13
We discussed methods for constructing analytic functions and, in particular, the convergence properties of power series. The assignment is shown on the last page of the slides from class.
Tuesday, September 11
We discussed applications of the Cauchy–Riemann equations to the range of analytic functions and to the simplest version of Cauchy’s integral theorem. The assignment is shown on the last page of the slides from class.
Thursday, September 6
We discussed the relationship between real differentiability and complex differentiability as evidenced by the Cauchy–Riemann equations. The assignment is shown on the last page of the slides from class.
Tuesday, September 4
We discussed the notions of limits of complex functions, continuity of complex functions, and derivatives of complex functions. The assignment is shown on the last page of the slides from class.
Thursday, August 30
We looked at an example of limits and discussed the notions of the “point at infinity” and stereographic projection. The assignment is shown on the last page of the slides from class.
Tuesday, August 28
At the first class meeting, we discussed definitions of the complex numbers and structures supported by the complex numbers. We worked in groups on an exercise about the geometry of the complex plane. As shown on the last page of the slides from class, the assignment is to read sections 1, 2, and 3 in Chapter I of the textbook.
Wednesday, August 1
This site went live today. Once the Fall 2018 semester starts, there will be regular updates about assignments and the highlights of each class meeting.