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Math 246a, Homework 9
Christoph Thiele
0.1 Paley’s theorem
From a qualifying exam: Let f and g be continuous integrable functions on the real line related by the Fourier transform,
f (x) =
∫
R
g(k)e−^2 πixk^ dk
g(k) =
∫ R
f (x)e^2 πixk^ dx
Prove that it is impossible that both f and g are compactly supported. Hint: Consider holomorphic extensions.
0.2 Conformal maps
From a qualifying exam: Find an explicit conformal mapping from the region
{|z| < 1 } \ [0, 1)
onto the upper half plane {Im(z) > 0 }. Hint: it is a composition of simpler maps such as power functions z → zα^ and linear fractional transformations.
0.3 More conformal maps
From a qualifying exam Let D be the domain in the complex plane C that is the intersection of the two open disks centered at ±1 whose boundary circles passes through ±i. Find a conformal map f of D onto the open unit disk ∆ = {|w| < 1 } such that f (i) = 1 and f (−i) = −1. You may express f as composition of other specific maps. What are the images of arcs of circles passing through ±i under your map? Justify your answer.
0.4 Yet more conformal maps
From a qualifying exam: Let f be any conformal mapping from the strip S = {z ∈ C : − 1 < =(z) < 1 } onto the unit disc D = {z : |z| < 1 } such that uniformly in y ∈ (− 1 , 1),
zlim→∞ f^ (x^ +^ iy) = 1
z→−∞lim f^ (x^ +^ iy) =^ −^1
Find the images in D of the set of horizontal lines in S and the set of vertical lines in S. Hint: in each case the set of images does not depend on the choice of f.
0.5 Riemann zeta
Find an expression for (^) ∑ (−1)nn−z
involving the zeta function. Discuss the region where this limit converges condi- tionally. Turn this into a proof that the Riemann zeta function has a meromorphic extension to <(z) > 0.
0.6 Modular function
Consider the open set civen by D/(D 1 ∪ D 2 ∪ D 3 where D is the open disc of radius one about 0, while Dj is the closed disc of radius 1 about ωj^ wheer ω is a third root of unity. a) Prove that there is a bijective conformal map from this region to the open unit disc. (use a famous theorem). b) Prove that this map can be analytically continued across the open circular boundary arcs (consider the locally harmonic function log |f (z)| and use Schwartz reflection principle. c) Prove that this map extends to a continuous bijection of the closure of the region to the closed disc, and let z 1 , z 2 , z 3 denote the image of the three corners. (The hard part is to discuss the behaviour near the corners. It helps to show that the gradient of log |f | (which is related to the derivative of f ) times the square of the distance to the corner tends to zero as one approaches the corner by using the mean value theorem to express the gradient of the function.) d) Normalize so that the corners are fixed under the map. Show that this map can be extended to a map form the open disc to the three times punctured Riemann sphere S \ {z 1 , z 2 , z 3 }. (Continued Schwartz reflection)
