Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 7E (Further Complex Methods)

The substitution $t\mapsto 1-t$ gives $B(p,q)=B(q,p)$. Direct integration gives

Integration by parts, or the identities obtained by splitting $1=t+(1-t)$, givesTherefore

For any $z\notin \{0,-1,-2,\dots \}$, choose $N$ with $\mathrm{Re}(z+N)>0$ and defineThe recurrence makes this independent of $N$ and agrees with the original integral where $\mathrm{Re}z>0$. It is the desired analytic continuation away from the listed points.

Usingwe see that $z=-n$ is generally a simple pole, inherited from $\Gamma (z)$. A cancellation occurs exactly when $p-n$ is a nonpositive integer. Thus $z=0$ is always a simple pole; for $n\ge 1$, $z=-n$ is removable when $p$ is one of $1,2,\dots ,n$, and otherwise is a simple pole.