Math 213a: Complex analysis Problem Set #6 (29 October 2003): The Gamma function; univalent functions and normal families

1. [Gauss multiplication formula] Let n be a positive integer, and define

F (z) = ∏_{k=0}^{n−1} Γ(\frac{z + k}{n} ).

i) Show that F (z) has the same poles as Γ(z), and satisfies the functional equation F (z + 1) = zF (z)/n.

ii) This suggests that F (z) should be proportional to n^{−z}Γ(z). Prove that this is in fact the case, and determine the constant of proportionality.

2. Determine for each n = 0, 1, 2, … the residue of Γ(z) at the pole z = −n. Use this to compute ∫_{γ} Γ(s)x^{s} ds for all x > 0, where γ is the contour {1 + it | t ∈ R}.

3. Prove the integral formula

∫_{−∞}^{∞} |Γ(σ + it)|^{2}a^{it} dt = 2πΓ(2σ)\frac{a^{σ}}{(1 + a)^{2σ}}

for all real and positive a, σ. For which complex a does this formula remain valid?

This result, together with the inversion formula for Fourier transforms, yields a closed form for the Fourier transform of (sech x)^{2σ}; In particular, for σ = 1/2 we recover the formula for the Fourier transform of sech(x). For general σ > 0 we can also obtain the orthogonal polynomials for the weight function |Γ(σ + it)|^{2} — that is, polynomials P_{n}(t) of degree n in t such that

I(m, n) := ∫_{−∞}^{∞} |Γ(σ + it)|^{2}P_{m}(t)P_{n}(t)dt

vanishes unless m = n. One nice way is to use the coefficients in the generating function

(1 + ix)^{−σ−it}(1 − ix)^{−σ+it} = ∑_{n=0}^{∞} P_{n}(t)x^{n}.

Can you determine I(n, n), or more generally

∫_{−∞}^{∞} |Γ(σ + it)|^{2}P_{l}(t)P_{m}(t)P_{n}(t)dt

for nonnegative integers l, m, n? These P_{n}(t) are the symmetric Meixner-Pollaczek polynomials (“symmetric” because one can more generally describe the orthogonal polynomials for the weight function |Γ(σ + it)|^{2}e^{ct} for any constant c with |c| < π/2).

Back to complex analysis: the remaining problems concern normal families and univalent functions.

4. Let

f(z) = \frac{1}{z} + ∑_{n=0}^{∞} a_{n}z^{n}

be a \underline{univalent} meromorphic function on the open unit disc Δ = \{|z| < 1\}. Show that for each positive r < 1 the complement in C of {f(z): |z| < r} has area

π(\frac{1}{r^{2}} − ∑_{n=0}^{∞} n|a_{n}|^{2}r^{2n}),

and thus that the complement of f(Δ) has area

π(1 − ∑_{n=0}^{∞} n|a_{n}|^{2}).

[Recall from the zeroth problem set that the area enclosed by a simple closed analytic arc γ is (1/2i) ∮_{γ} z̄dz.] Conclude that the univalent functions 1/z + O(z) on Δ constitute a normal family.

5. Let F be the family of analytic functions f(·) on Δ such that f is univalent

and normalized by f(0) = 0 and f′(0) = 1.

i) Show that if f ∈ F then F also contains a function g(·) such that (g(z))^{2} = f(z^{2}) for all z ∈ Δ.

ii) Apply the result of Problem 4 to conclude that F is a normal family.

In particular it follows that for each n there is an upper bound B_{n} on the absolute value of the z^{n} coefficient in the Taylor expansion of any f ∈ F. The Bieberbach conjecture, finally proved in the mid-1980’s by L. de Branges, asserts that the least bound is n, and specifies all functions attaining this bound. The next problem gives an easy first step in this direction, and some more information about F.

6. i) Show that |f′′(0)| ≤ 4 for all f ∈ F, with equality if and only if f(z) = z/(1 − cz)^{2} for some c ∈ C with |c| = 1. [Such f turn out to be the only functions attaining the Bieberbach bound for any n.]

ii) Prove that for each f ∈ F there exists w ∈ C such that w ∉ f(Δ) and |w| ≤ 1.

This problem set is due Wednesday, November 5, at the beginning of class.