Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 8D (Further Complex Methods)

Table of contents
- Solution 8D

Solution

The two Gamma function integrals are absolutely convergent when

Re α, Re β > 0

. Consequently Fubini theorem gives

Γ (α) Γ (β) = \int_{0}^{\infty} \int_{0}^{\infty} x^{α - 1} y^{β - 1} e^{- (x + y)} d x d y .

Make the change of variables

u = x + y

,

t = x / (x + y)

, so

x = u t

,

y = u (1 - t)

,

u > 0

,

0 < t < 1

, and the Jacobian is

u

. Powers of positive real variables use the real logarithm, so this transformation is also valid for complex exponents. The integral factors as

[\int_{0}^{\infty} u^{α + β - 1} e^{- u} d u] [\int_{0}^{1} t^{α - 1} (1 - t)^{β - 1} d t] .

Both factors converge under the same real-part assumptions. The first is

Γ (α + β)

, proving the Beta function identity

Γ (α) Γ (β) = Γ (α + β) \int_{0}^{1} t^{α - 1} (1 - t)^{β - 1} d t .

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