 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/ii/paper-4/8b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

For the Euler integral to converge directly, assume

Re b > 0

and

Re (c - b) > 0

. At zero it is the beta function

B (b, c - b) = Γ (b) Γ (c - b) /Γ (c)

. The normalization is therefore

K = \frac{Γ ( c )}{Γ ( b ) Γ ( c - b )} .

Outside these initial parameter conditions the hypergeometric function is defined by analytic continuation wherever the parameters permit it.

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