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Math 246a, Homework 8
Christoph Thiele
0.1 Modular function
There exists a holomorphic function λ from the upper half plane {z : =(z) > 0 } onto the set C \ { 0 , 1 } (“triple punctured Riemann sphere” whose derivative vanishes nowhere. A famous such function is called the modular function). Prove that every entire function f omitting the values 0 and 1 is constant. (Hint: Prove there is a g such that f = λ ◦ g.)
0.2 Doubly periodic functions
- Let f be an entire function such that
f (z) = f (z + 1) = f (z + i)
for all z ∈ R^2. Prove that f is constant.
- Let f be a meromorphic function such that
f (z) = f (z + 1) = f (z + i)
for all z ∈ R^2 and f has at most simple poles at all points x + iy with x, y integers and no other poles. (We will denote the set of points with integer coordinates by Z^2 ) Prove that f is holomorphic. Hint: calculate the residue of f at (0, 0) by integrat- ing along a square with corners (± 12 , ±^12 ).
0.3 More doubly periodic functions
- Prove that the sequence of functions
fN (z) =
∑^ N
n=−N
∑N
k=−N
(z − (n + ik))^3
converges uniformly on compact sets in Ω = R^2 \ Z^2 to a non-constant holomorphic function P 3 in Ω, which has a meromorphic extension to all of R^2.
- Prove that the sequence of functions
fN (z) =
z^2
∑
−N ≤n,k≤N,(k,n) 6 =(0,0)
(z − (n + ik))^2
(n + ik)^2
converges uniformly on compact sets in Ω = R^2 \ Z^2 to a non-constant holomorphic function P 2 in Ω, which has a meromorphic extension to all of R^2.
0.4 Yet more doubly periodic
- Prove that every doubly periodic meromorphic function (with respect to 1 and i) which is holomorphic outside Z^2 is a polynomial in P 2 and P 3 constructed above. (Hint: discuss the principal part at 0)
- Prove that P 32 is a cubic polynomial in P 2
0.5 Yet Yet more doubly periodic
- Prove that for each doubly periodic function (w.r.t. 1, i), the number of zeros and number of poles, counted with multiplicities, are equal.
- Prove that every doubly periodic meromorphic function (with respect to 1 and i) is a rational function in P 2 and P 3. (Hint: split each such function into even and odd part about 0. and then start dividing by linear factors in P 2 to reduce the degree.)
0.6 Normal families
Prove the following theorem: Let (fn) be a sequence of holomorphic functions on the open unit disc D such that supz∈D |fn(z)| ≤ 1 for all n. Then there is a subsequence of this sequence which converges uniformly on compact sets. Hint: Pick a sequence of points z 1 , z 2 ,... which are dense in D. By Heine Borel there exists an increasing sequence nm of indices such that limm→∞ fnm (z 1 ) exists. Write (fn, 1 ) for the subsequence fm, 1 := fnm of (fn). Inductively we can choose subsequences fn,k of fn,k− 1 which converge ot zk. Then the sequence gn = fn,n converges at all z 1 , z 2 ,.. .. To prove that gn converges uniformly on compact sets in D, write for z in a compact set K ⊂ D
gn(z) = gn(zk) +
∫
γ
g′(z)dz
where zk is near (to be made precise) z and γ is a straight line from zk to z. Use a good bound for g′(z) on the compact set K from the Cauchy integral formula.
