= Argument principle for a rational function
{title2=$f(z)=\dfrac{(z-1/4)^2(z-2)}{z+1/2}$}
Consider the <rational function>
$$
f(z)=\frac{(z-1/4)^2(z-2)}{z+1/2}.
$$
Inside the <unit circle> are a double <zero of a function> at $1/4$ and a simple <pole> at $-1/2$. The simple <zero of a function> at $2$ is outside. Thus \b[$N=2$, $P=1$, and the <winding number> is one]. The factorization checks the count exactly; the plot illustrates how the boundary records it.
\Image[/argument-principle-rational-boundary.png]
{title=A double zero and a simple pole leave one net turn of the boundary image}
{description=Left: the positively oriented <unit circle> surrounds the double <zero of a function> at $1/4$ and simple <pole> at $-1/2$. Right: the <image> <contour> under the stipulated <rational function>, with matching colored boundary points and arrows. Its net <winding number> about zero is one. The curve is computed from the formula, not copied from a source.}
For the concentric <circle> $|z|=r$, the <winding number> is zero for $0<r<1/4$, two for $1/4<r<1/2$, one for $1/2<r<2$, and two for $r>2$. At $r=1/4$ or $r=2$, the boundary contains a <zero of a function>; at $r=1/2$, it contains a <pole>. The theorem does not apply at those three radii. Moving a boundary changes the count only when it crosses an object being counted; <homotopy invariance of winding number> describes the unchanged count between crossings.
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