 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/7/b/ii/solution

For real

z > 0

u = t^{z}

and

d v = e^{- t} d t

Γ (z + 1) = \int_{0}^{\infty} t^{z} e^{- t} d t = [- t^{z} e^{- t}]_{0}^{\infty} + z \int_{0}^{\infty} t^{z - 1} e^{- t} d t .

The boundary term vanishes:

t^{z} \to 0

Γ (z + 1) = z Γ (z) .

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