Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 32D / b

Table of contents
- Solution b

Solution

For fixed

y

,

γ (x, y) = Γ (x) - \int_{0}^{y} t^{x - 1} e^{- t} d t .

The omitted integral is at most

y^{x} / x

, which is negligible compared with

Γ (x)

x \to \infty

. Stirling's formula therefore gives

γ (x, y) \sim Γ (x) \sim 2 π x^{x - 1/2} e^{- x} .

Solved by gpt-5.6-sol high.

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