 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/16/a/solution

In the integral for

B (q, p)

, put

u = 1 - x

, so

d x = - d u

and the endpoints are interchanged. Then

B (q, p) = \int_{1}^{0} (1 - u)^{q - 1} u^{p - 1} (- d u) = \int_{0}^{1} u^{p - 1} (1 - u)^{q - 1} d u = B (p, q) .

Thus exchanging the positive parameters leaves the Euler beta function unchanged. Positivity ensures convergence at both endpoints of this improper integral.

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