Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 16 / d

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- Solution d

Solution

Multiply the two defining Gamma integrals. For

p, q > 0

the nonnegative product integrand allows Tonelli's theorem:

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

Use

r = x + y

,

u = x / (x + y)

, so

x = r u

,

y = r (1 - u)

and the positive quadrant becomes

r > 0

,

0 < u < 1

. The Jacobian determinant has absolute value

det (u 1 - u r - r) = r .

After including this factor, the transformed integral separates:

[\int_{0}^{\infty} r^{p + q - 1} e^{- r} d r] [\int_{0}^{1} u^{p - 1} (1 - u)^{q - 1} d u] = Γ (p + q) B (p, q) .

This proves the beta-gamma identity,

Γ (p) Γ (q) = B (p, q) Γ (p + q) .

All changes of variables are made inside the open positive domains; their boundaries do not contribute to the integrals.

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