 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/7/ii/a/solution

For

Re z > 0

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 because

t^{Re z} \to 0

at zero and exponential decay dominates any fixed power at infinity. Thus

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

This also gives

Γ (n + 1) = n!

for nonnegative integers, linking the Gamma function to the factorial.

 New to topics? Read the docs here!