# Topics

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

## Review

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### Integrals and Residues

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Bounds

Jordan’s Lemma:

# 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 $$f(z) = \sum_{n=0}^\infty c_n(z-z_0)^n$$ 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

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#qualifying_exam #active_projects