= Argument principle
{title2=$N-P=\operatorname{wind}(f\circ\gamma,0)$}
{wiki}
The argument principle turns a boundary journey into an interior count: follow the values of a <meromorphic function> around a <contour>, and count their net turns around zero. Every enclosed <zero of a function> contributes its <multiplicity of a zero>; every enclosed <pole> subtracts its order. There is no need to find each <zero of a function> separately.
Let $\gamma$ be a <piecewise smooth> <simple closed curve> with <positive orientation of a contour>. Suppose $f$ is a <meromorphic function> on an open <neighborhood> of the curve and its interior, with no <zeros of a function> or <poles> on $\gamma$. Then
$$
\boxed{N-P=\frac1{2\pi i}\oint_\gamma\frac{f'(z)}{f(z)}\,dz=\operatorname{wind}(f\circ\gamma,0)}.
$$
Here $N$ is the sum of the <zero multiplicities> inside $\gamma$, and $P$ is the sum of the orders of its enclosed <poles>. The <image> <contour> $f\circ\gamma$ may cross itself or revisit a point. Its signed <winding number>, rather than the number of visually bounded regions, is the quantity that counts.<Topic 11: Argument Principle by Jeremy Orloff>[§11.2, Theorems 11.1 and 11.4, pp. 1–3]{ref}
For the <polynomial> $f(z)=z^3$ and the <unit circle> $\gamma(t)=e^{it}$, $0\leq t\leq2\pi$, the <image> is $e^{3it}$. It goes around zero three times although its drawn outline is just one circle. The interior contains one distinct <zero of a function>, at zero, with <multiplicity of a zero> three: \b[the count is three]. Reversing the <orientation of a contour> reverses the <winding number> and the sign of the <contour integral>.
An <argument> must be followed continuously along the journey. Subtracting the principal <arguments> at the endpoints loses whole turns: a closed <image> <contour> has the same endpoint value even when its <winding number> is nonzero. Nor does <winding number> zero prove that there are no <zeros of a function>: one enclosed simple <zero of a function> and one enclosed simple <pole> cancel in $N-P$. Known <poles> must be added back to recover $N$.
The mechanism is short. Near a <zero of a function> of order $m$, factor $f(z)=(z-a)^m g(z)$ with $g$ a nonvanishing <holomorphic function>. Its <logarithmic derivative> is $f'/f=m/(z-a)+g'/g$, so its <residue> is $m$. A <pole> of order $m$ gives <residue> $-m$. The <residue theorem> adds those contributions. Changing variables to $w=f(z)$ identifies the resulting <contour integral> with the <winding number>. This is a local algebraic count joined to a global geometric count.<Topic 11: Argument Principle by Jeremy Orloff>{ref}
The usefulness goes beyond explicit <polynomials>. <Rouché's theorem> keeps a count unchanged when a boundary <perturbation> cannot push the <image> <contour> through zero. The <Nyquist stability criterion> applies the same counting idea to <poles> governing a system under <feedback control>.<Topic 11: Argument Principle by Jeremy Orloff>[§11.2.3 and §11.3.6]{ref}
Back to article page
