Tags: #qualifying_exam #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 Cauchy-Riemann 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, Cauchy-Riemann equations.

• 

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

• 

Exercises

Qual Problems

• Pasted image 20210517025502.png
• 
• 
• 
• 
• 
• 
• 
•