Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 7E (Further Complex Methods)

Table of contents
- Solution 7E

Solution

The given Euler product for the gamma function has the finite approximants

Γ_{m} (z) = \frac{m ! ( m + 1 ) ^{z}}{z ( z + 1 ) \dots ( z + m )}, Γ_{m} (z) ⟶ Γ (z)

away from the poles. Split the denominator of

Γ_{2 m + 1} (2 z)

into even and odd factors. The resulting exact identity is

\frac{Γ _{2 m + 1} ( 2 z )}{2 ^{2 z} Γ _{m} ( z ) Γ _{m} ( z + \frac{1}{2} )} = \frac{( 2 m + 1 )!}{2 ^{2 m + 2} ( m ! ) ^{2} m + 1} .

The right side does not depend on

z

, so its limit is a constant

C

. Evaluate the ratio at

z = 1/2

:

Γ (1) = 1

, and the gamma reflection formula, together with positivity on the positive real axis, gives

Γ (1/2) = π

. Hence

C = \frac{1}{2 π}, Γ (2 z) = \frac{2 ^{2 z - 1}}{π} Γ (z) Γ (z + \frac{1}{2}) .

This is the gamma duplication formula. The argument first applies where the factors are finite; the identity then holds as an identity of meromorphic functions, with values at poles interpreted accordingly.

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