# 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.png

### Integrals 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.png

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

Pasted image 20210527173251.png    #qualifying_exam #active_projects