= Cancellation of positive-integer gamma singularities on a Hankel contour
Keep the circular portion of a <Hankel contour> at fixed nonzero radius. Its numerator $I(z)=\int_Ht^{z-1}e^t\,dt$ is an <entire function> of $z$, because the contour avoids zero and its tails decay exponentially, uniformly on compact parameter sets. At a positive integer $n$, the integrand is single-valued and entire in $t$, so the bank integrals cancel and the circular integral is zero. Thus $I(n)=0$, canceling the simple zero of $2i\sin(\pi z)$ in the Hankel representation of the <Gamma function>. At a nonpositive integer $-n$, the circular integral instead gives $2\pi i/n!$, yielding residue $(-1)^n/n!$.
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