Tags: #qualifying_exam #active_projects

Topics

• [Blaschke factors\]
• Toy contours
• Cauchy’s integral formula
• Cauchy inequalities
• Computing integrals
  • Residue formulas
  • ML Inequality
  • Jordan’s lemma

Review

Integrals and Residues

Residues

Bounds

Jordan’s Lemma: \_attachments/Pasted image 20210527182026.png

Blaschke Factors

Cauchy’s Integral Formula

Misc

Warmups

• Do any example from here

• \_attachments/Pasted image 20210527174041.png

• 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.

• \_attachments/Pasted image 20210527175840.png
• \_attachments/Pasted image 20210527180104.png

Questions

• Can every continuous function on \(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu\) 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 \({\mathbb{C}}\), and for each \(z_0 \in {\mathbb{C}}\) there is at least one coefficient in the expansion \(f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n\) 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