Tags: #active_projects
Topics
 [Blaschke factors\]
 Toy contours
 Cauchy’s integral formula
 Cauchy inequalities

Computing integrals
 Residue formulas
 ML Inequality
 Jordan’s lemma
Review
Obsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527180305.pngObsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527180305.pngIntegrals and Residues
Obsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527181024.pngObsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527175221.pngObsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527175221.pngObsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527175221.pngObsidian/Workshops/Complex Analysis/_attachments/Pasted image 20210527175221.pngResidues
Bounds
Jordan’s Lemma:
Blaschke Factors
Cauchy’s Integral Formula
Misc
Warmups

Do any example from here


Anything from the homeworks

Show that \(f'=0 \implies f\) is constant using integrals and primitives (i.e. antiderivatives).
See S&S Corollary 3.4.
Questions
 Can every continuous function on \(\overline{{\mathbb{D}}}\) be uniformly approximated by polynomials in the variable \(z\)?
Hint: compare to Weierstrass for the real interval.
 Suppose \(f\) is analytic, defined on all of \({\mathbf{C}}\), and for each \(z_0 \in {\mathbf{C}}\) there is at least one coefficient in the expansion \(f(z) = \sum_{n=0}^\infty c_n(zz_0)^n\) is zero. Prove that \(f\) is a polynomial.
Hint: use the fact that \(c_n n! = f^{(n)}(z_0)\) and use a countability argument.
Qual Problems
Pasted image 20210527173251.png