Complex Analysis

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  • Given a function continuous in a disk and analytic everwhere but at the center, prove that the function is analytic in the entire disk.
  • Give a proof of Picard's theorem using, for example, the fact that the j invariant of a modular curve uniformizes the 2,3,\infty hyperbolic triangle as the upper half-plane.
  • Show that the mapping group of the torus is SL(2,\Z).
  • Let \Omega={\mathbb C}\backslash\left\{x\in \R\mid x<{1\over 4}\right\}. Is there a conformal isomorphism f\colon \Delta\to\Omega, where Δ is the open unit disk? (McMullen)
  • Is there one with f(0) = 0? (McMullen)
  • How can we arrange for a unique f with f(0) = 0? (McMullen)
  • What can you say about the coefficients ai of the power series expansion f=\sum a_jz^j? (McMullen)
  • For f with f'(0)\in \R, f'(0)\geq 0, what ring do the ai lie in? (McMullen)
  • So to show that the ai lie in this ring, can we write down another function in terms of f and --'s which maps Δ to Ω? (McMullen)
  • Calculate ai. Now what ring do the ai lie in? (McMullen)
  • How would you write down the power series for tanz? (McMullen)
    • What is its radius of convergence? (McMullen)
  • Can you prove what the zeroes of cosz are? (McMullen)
    • Why does the radius of convergence correspond this way? (McMullen)
  • What is the area of a spherical triangle? Can you prove it? (McMullen)
  • Same for hyperbolic triangle. (McMullen)
  • Define a complex torus. (McMullen)
  • What is the automorphism group of a complex torus? (McMullen)
  • Show that if all the zeroes of a polynomial lie in a half-plane, then all zeroes of the derivative lie in the same half plane.
  • What is the area of a spherical triangle?
  • What is the automorphism group of a complex torus?
  • If fn is a family of holomorphic functions such that f_n\to F uniformly on compact subsets of some domain Ω, what can you say about fn'?
  • Give an example of a sequence f_n\to f where every fn is holomorphic and injective, and f is not. Is this the most general such example?
  • Why is there no conformal automorphism from the punctured disk to an annulus?
  • Show that for a doubly periodic function f the number of zeroes of f and the number of poles of f (counting with multiplicities) is equal.
  • Suppose fi are harmonic functions on the unit disk D. Show that no linear combination of the fi can be negative on \partial D and positive at some point in the interior of D.
  • Find the poles and residues of 1 / sin(z).
  • Give the formula for a conformal map from the unit disk to the inside of a polygon with angles 2π − βiπ.
  • Show that a continuous real-valued function u on some region Ω which has the mean-value property is harmonic.
  • Suppose f is an analytic map from the punctured disk to {\mathbb C}. Can you write a power series expansion for f? What general form does it have? (McMullen)
  • What can you say about the growth of the an's?
  • Relate this to radii of convergence.
  • If f is bounded, what additional things can you say?
  • How would you compute the integral \int_0^{\infty}{x^{1\over 2}\over {1+x^2}}\,dx?(McMullen)
  • Is the top half of a disk conformally isomorphic to the whole disk? What is the isomoprhism? (McMullen)
  • What is the argument principle? (Sarason)
  • Why is it called the argument principle? (Sarason)
  • Generalize the argument principle to a statement about an arbitrary continuous function f from a domain to {\mathbb C}. You can assume f has isolated zeroes. (Sarason)
  • You don't want to calculate any integrals do you? (Sarason)
  • How do you prove the uniqueness part of the Riemann Mapping Theorem? (Sarason)
  • What are the conformal automorphisms of the disk? (Sarason)
  • What are the conformal automorphisms of the upper half plane? (Sarason)
  • What is the modular group? (Sarason)
  • Let f be holomorphic in Δ * , the punctured unit disk, and suppose that |f(z)| \le {1 \over {\sqrt {|z|}}}. Show that the singularity at 0 is removable. (McMullen)
  • What is the Riemann zeta function? (Poonen)
  • State the Riemann hypothesis. (Poonen)
  • What is analytic continuation? Why is it unique?
  • Does every complex analytic function have a power series?
  • What is a a holomorphic function? What have properties do they have? (Sarason)
  • How can you prove they have a Taylor series expansion? (Sarason)
  • What connection does complex analysis have to algebraic geometry? (Sarason)

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