Math 213a: Complex analysis
Problem Set #6 (29 October 2003):
The Gamma function; univalent functions and normal families
1. [Gauss multiplication formula] Let nbe a positive integer, and define
F(z) =
n−1
Y
k=0
Γz+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−zΓ(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 RγΓ(s)xsds for all x > 0, where γis the contour
{1 + it |t∈R}.
3. Prove the integral formula
Z∞
−∞
|Γ(σ+it)|2ait dt = 2πΓ(2σ)aσ
(1 + a)2σ
for all real and positive a, σ. For which complex adoes 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
Pn(t) of degree nin tsuch that
I(m, n) := Z∞
−∞
|Γ(σ+it)|2Pm(t)Pn(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 =
∞
X
n=0
Pn(t)xn.
Can you determine I(n, n), or more generally
Z∞
−∞
|Γ(σ+it)|2Pl(t)Pm(t)Pn(t)dt
for nonnegative integers l, m, n? These Pn(t) are the symmetric Meixner-Pollaczek
polynomials (“symmetric” because one can more generally describe the orthogonal
polynomials for the weight function |Γ(σ+it)|2ect for any constant cwith |c|< π/2).
