Intro Complex Analysis, Lec 27, Review Cauchy's Theorem, Cauchy Integral Formulas, and Corollaries

Lecture 27. (0:00) Journal catch-up time and Mathematica project time. (1:08) This lecture will mostly be a review of what we've done since the last exam to try to get these ideas organized and solidified in our minds. (2:00) Write out the statement of Cauchy's Theorem. (7:11) Verbal description of continuous deformation theorem. (8:58) Write out the statement of the Cauchy integral formula (as a theorem). This can help you evaluate integrals. It is also a theoretical tool, such as to prove Liouville's theorem and facts about sequences and series. (15:29) Lemma about creating an analytic function over the the entire complex plane minus a contour and how to differentiate it (by differentiating inside the integral sign). (20:31) This lemma is useful for proving that analytic functions are infinitely differentiable (this is not true for functions that are merely differentiable on the real line). (23:32) Statement of the generalized Cauchy integral formula. This is once again useful for evaluating integrals. This is also useful for proving bounds about derivatives which leads to a proof of Liouville's Theorem. (29:58) Statement of Liouville's Theorem: Any entire function that is bounded must be constant. (So if an entire function is not constant, it is not bounded). (33:00) Go over a homework exercise: if f(z) is an entire function and the real part Re(f(z)) is bounded above on the entire complex plane, then f(z) is constant. (36:37) Statement of the maximum modulus principle in two forms, for general domains and for bounded domains. Note: The modulus of a nonconstant analytic function would go to infinity if we assume the domain is UNbounded. (41:26) Reminder of the example of a fifth degree polynomial function and the fact that there is a maximum principle for harmonic functions (those that solve Laplace's equation). Therefore, critical points of harmonic functions must be saddle points. Harmonic functions turn out to be infinitely differentiable as well. (48:47) Section about applying complex analysis to studying harmonic functions (solutions of Laplace's equation). For example, the Poisson integral formula is a way of representing a harmonic function on a disk. (50:10) The form of the Taylor series of a complex function f(z) that is analytic at a point z0. This can be computed because f will be infinitely differentiable at z0. It will also equal f(z) in a neighborhood of z0 (the disk of convergence, with a radius of convergence, which could be infinity). This does not work for real infinitely differentiable functions. The convergence will be uniform inside the disk of convergence, which means you can differentiate and integrate term by term.

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