Complex Analysis Qual Prep Week 1: Preliminaries

Tags: #qualifying_exam #active_projects #qual_complex_analysis

Week 1: Preliminaries


  • 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


  • 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.

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    • Use that Lipschitz implies uniformly continuous.
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    • Derivation of CR equations: approach along totally real and totally imaginary paths for \(h\).
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    • Polar coordinates, chain rule, Cauchy-Riemann equations.
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    • Converges everywhere on \(S^1\): take \(\sum z^k/k^2\).
    • Part 2: ???? Todo get help
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Qual Problems

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#qualifying_exam #active_projects #qual_complex_analysis