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Homework Assigned on April 16, 2009 due April 21, 2009
Problem 1 (from Stein & Shakarchi, p.254, #24). The elliptic integrals K and K′^ defined by 0 < k < 1 by
K(k) =
0
dx ((1 − x^2 )(1 − k^2 x^2 ))
1 2
K′(k) =
k 1
dx ((x^2 − 1)(1 − k^2 x^2 ))
(^12)
satisfy various interesting identities. For instance.
(a) Show that if ˜k^2 = 1 − k^2 and 0 < k < 1, then
K′^ (k) = K
˜k
[Hint: Change variables x =
1 − ˜k^2 y^2
in the integral defining K′(k).]
(b) Show that if k˜^2 = 1 − k^2 and 0 < k <˜ 1, then
K(k) =
1 + ˜k
K
1 − ˜k 1 + k˜
Hint: Change variables
x =
2 t 1 + ˜k + (1 − ˜k)t^2
(c) For α, β ∈ C and γ 6 = 0, − 1 , − 2 , · · · the hypergeometric series is defined by
F (α, β, γ; z) = 1 +
∑^ ∞
n=
α(α + 1) · · · (α + n − 1)β(β + 1) · · · (β + n − 1) n!γ(γ + 1) · · · (γ + n − 1)
zn.
Verify that
F (α, β, γ; z) =
Γ(γ) Γ(β)Γ(γ − β)
0
tβ−^1 (1 − t)γ−β−^1 (1 − zt)αdt
for α > 0 , β > 0 , γ > β and |z| < 1. Show that for 0 < k < 1 one has
K(k) =
π 2
F
, 1; k^2
Problem 2 (from Stein & Shakarchi, p.256, #3). The Schwarz-Pick lemma is the infinitesimal version of an important observation in complex analysis and geometry. For complex numbers w ∈ C and z ∈ D define the hyperbolic length of w at z by
‖w‖z =
|w| 1 − |z|^2
where |w| and |z| denote the usual absolute values. This length is sometimes referred to as the Poincar´e metric, and as a Riemannian metric it is written as
ds^2 =
|dz|^2 (1 − |z|^2 )^2
The idea is to think of w as a vector lying in the tangent space at z. Observe that for a fixed w, its hyperbolic length grows to infinity as z approaches the boundary of the disk. We pass from the infinitesimal hyperbolic length of tangent vectors to the global hyperbolic distance between two points by integration.
(a) Given two complex numbers z 1 and z 2 in the disk, we define the hyperbolic distance between them by
d (z 1 , z 2 ) = inf γ
0
‖γ′(t)‖γ(t) dt,
where the infimum is taken over all smooth curves γ : [0, 1] → D joining z 1 and z 2. Use the Schwarz-Pick lemma to prove that if f : D → D is holomorphic, then
d (f (z 1 ) , f (z 2 )) ≤ d (z 1 , z 2 ) for z 1 , z 2 ∈ D.
In other words, holomorphic functions are distance-decreasing in the hyper- bolic metric.
(b) Prove that automorphisms of the unit disk preserve the hyperbolic dis- tance, namely,
d (ϕ (z 1 ) , ϕ (z 2 )) ≤ d (z 1 , z 2 ) for z 1 , z 2 ∈ D
The inner radius of a region K ⊂ D that contains the origin is defined by
rK = sup
ρ ≥ 0
∣ D(0, ρ) ⊂ K
where D(0, ρ) is the open disk centered at 0 with radius ρ. Also, a holo- morphic injection f : K → D is said to be an expansion if f (0) = 0 and |f (z)| > |z| for all z ∈ K − { 0 }.
(a) Prove that if f is an expansion, then rf (K) ≥ rK and |f ′(0)| > 1. [Hint: Write f (z) = zg(z) and use the maximum principle to prove that |f ′(0)| = |g(0)| > 1.]
(b) Suppose we begin with a Koebe domain K 0 and a sequence of expansions {f 0 , f 1 , · · · , fn, · · · }, so that Kn+1 = fn (Kn) are also Koebe domains. We then define holomorphic maps Fn : K 0 → D by Fn = fn ◦ · · · ◦ f 0. Prove that for each n, the function Fn is an expansion. Moreover, F (^) n′(0) =
∏n k=0 f^
′ k(0), and conclude that limn→∞ |f (^) n′(0)| = 1. [Hint: Prove that the sequence {|F (^) n′(0)|} has a limit by showing that it is bounded above and monotone increasing. Use the Schwarz lemma.]
(c) Show that if the sequence is osculating, that is, rKn → 1 as n → ∞, then {Fn} converges uniformly on compact subsets of K 0 to a conformal map F : K 0 → D. [Hint: If rF (K 0 ) ≥ 1, then F is surjective.]
(d) To construct the desired osculating sequence we shall use the automor- phisms
ψα =
α − z 1 − αz ¯
Given a Koebe domain K, choose a point α ∈ D on the boundary of K such that |α| = rK, and also choose β ∈ D such that β^2 = α. Let S denote the square root of ψα on K such that S(0) = 0. Why is such a function well defined? Prove that the function f : K → D defined by f (z) = ψβ ◦ S ◦ ψα is an expansion. Moreover, show that
|f ′(0)| =
1 + rK 2
rK
[Hint: To prove that |f (z)| > |z| on K − { 0 } and apply the Schwarz lemma to the inverse function, namely ψα ◦ g ◦ ψβ where g(z) = z^2 .]
(e) Use Part(d) to construct the desired sequence.
