Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 13F / Solution

For a closed piecewise smooth path $\gamma$ avoiding $w$, its winding number isA meromorphic function has a zero of order $m>0$ at $z_0$ whenwith $g$ holomorphic; it has a pole of order $m$ at $w_0$ when $(z-w_0{)}^{m}f(z)$ extends holomorphically and nonvanishingly there.

The argument principle states that, if $f$ has no zero or pole on $\gamma$,For a positively oriented simple contour this is the number of enclosed zeros minus poles, counted with multiplicity. It follows from the residue theorem because the residue of $f^{\prime }/f$ at a zero of order $m$ is $m$, while at a pole of order $m$ it is $-m$.

LetOn the imaginary axis,Thus $p(iy)$ always lies in the open right half-plane, has no zero there, and its continuous argument has net change zero as the imaginary-axis part of a large right-half-disc contour is traversed. On the right semicircle, $p(z)/z^4\to 1$ uniformly, so the argument change tends to that of $z^4$, namely $4\pi$. The argument principle therefore givesroots in the open right half-plane, counted with multiplicity.