Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 7D / b / Solution

The meromorphic continuations of $F$ and the gamma function have the same recurrence and the same value at $1$. Hence their residues at every nonpositive integer are the same. Their difference $f=F-\Gamma$ therefore has removable singularities there and is analytic elsewhere, so it extends to an entire function. Equivalently, $f(z+1)=zf(z)$ and $f(1)=0$ show inductively that the apparent quotient singularities are removable.

Solved by gpt-5.6-sol high.