Tags: #active_projects #qual_complex_analysis
Week 1: Preliminaries
Topics
 Complex arithmetic and geometry, conic section equations
 Uniform (continuity, differentiability, convergence)
 Inverse and implicit function theorems
 Green’s theorem, Stokes theorem
 Complex plane, Riemann sphere
Warmup

State the CauchyRiemann equations.

Define what it means for a function to be
 Holomorphic
 Meromorphic
 Analytic
 Harmonic
 Uniformly continuous
 Uniformly bounded
 Entire

What does it mean for a sequence or series to uniformly converge?

State the Laplace equation.

What is the Dirichlet problem?

Discuss how to carry out partial fraction decomposition

Determine the radius of convergence of the power series for \(\sqrt z\) expanded at \(z_0= 4 + 3i\).

What is the logarithmic derivative?

Find a function \(f\) such that \(f^2\) is analytic on the open unit disc but \(f\) is not.

 Use that Lipschitz implies uniformly continuous.

 Derivation of CR equations: approach along totally real and totally imaginary paths for \(h\).

 Polar coordinates, chain rule, CauchyRiemann equations.

 Converges everywhere on \(S^1\): take \(\sum z^k/k^2\).
 Part 2: ???? Todo get help

Exercises
Qual Problems
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