Week 1: Preliminaries

Topics

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.
\_attachments/Pasted image 20210517025125.png
- Use that Lipschitz implies uniformly continuous.
\_attachments/Pasted image 20210517022935.png
- Derivation of CR equations: approach along totally real and totally imaginary paths for \(h\).
\_attachments/Pasted image 20210517022946.png
- Polar coordinates, chain rule, Cauchy-Riemann equations.
\_attachments/Pasted image 20210517023232.png
- Converges everywhere on \(S^1\): take \(\sum z^k/k^2\).
- Part 2: ???? Todo get help
\_attachments/Pasted image 20210517023424.png