# 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

## Qual Problems

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