Argument principle ($N-P=\mathrm{wind}(f\circ \gamma ,0)$)

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 $\gamma$. ThenHere $N$ 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.^[ref]

For the polynomial $f(z)=z^3$ and the unit circle $\gamma (t)=e^{it}$, $0\le t\le 2\pi$, 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: 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 $f^{\prime }/f=m/(z-a)+g^{\prime }/g$, 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.^[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.^[ref]

Consider the rational functionInside 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 $N=2$, $P=1$, and the winding number is one. The factorization checks the count exactly; the plot illustrates how the boundary records it.

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.

The boundary can recover more than a count. Under the hypotheses of the argument principle, let $h$ be a holomorphic function on a neighborhood of the contour and its interior. If the enclosed zeros of a function are $a_j$ with zero multiplicities $m_j$ and the enclosed poles are $b_k$ of orders $n_k$, thenMultiplying the local logarithmic derivative by $h$ changes its residue from $m_j$ to $m_{j}h(a_j)$, or from $-n_k$ to $-n_{k}h(b_k)$. The residue theorem then gives the identity. Choosing $h=1$ recovers the argument principle; choosing $h(z)=z$ gives the sum of the enclosed zero locations minus the sum of the enclosed pole locations, both counted with zero multiplicities or pole orders.^[ref]

For the rational function in argument principle for a rational function, the $h(z)=z$ integral is $2(1/4)-(-1/2)=1$. Adding the known pole location $-1/2$ back gives a zero-location sum of $1/2$. Dividing by the zero count two gives the arithmetic mean of the enclosed zeros, with multiplicity, as $1/4$. Choosing $h(z)=z^k$ similarly gives the difference between the power-sum symmetric polynomials evaluated on the enclosed zero and pole locations, repeating each location according to its multiplicity or order. These identities let boundary information constrain both the number and aggregate locations of enclosed zeros of a function.

A finite list of function values does not determine the winding number of the intervening image contour. For example, sample $\gamma (t)=e^{it}$ only at $t_j=2\pi j/4$, $j=0,1,2,3$. The polynomials $f(z)=z$ and $g(z)=z^5$ agree at all four sampled points, because $\gamma (t_j{)}^4=1$. Joining those values by straight segments draws the same square for both. Yet their true winding numbers are one and five, and their enclosed zero multiplicities differ by four.

Refining samples can help resolve a given example, but an asserted root count needs a bound controlling what happens between them. One sufficient certificate is to enclose the image of each boundary subarc and its interpolating segment in a convex set that excludes zero. Straight interpolation then gives a homotopy avoiding zero, so the polygon and true image contour have the same winding number. A plot that merely looks closed and avoids zero at its sampled points is not that certificate.

If $p(z(t))=t$ and $p^{\prime }(z)\ne 0$, implicit differentiation gives $z^{\prime }(t)=1/p^{\prime }(z)$. At a crossing of the unit circle, the displayed radial derivative determines whether the root enters or leaves the disk as the real parameter increases. Root counts are constant between such boundary crossings, including multiplicities. At the crossing parameter itself, boundary roots are excluded from an open-disk count. This complements the winding number form of the argument principle and handles a self-intersecting image contour without guessing its enclosed regions.

Suppose a monic real quartic polynomial has no nonnegative real zero and its real part on the imaginary axis is everywhere positive. For sufficiently large $R$, its image of the quarter-circle arc from $R$ to $iR$, closed by the straight chord from $p(iR)$ to $p(R)$, has winding number one: compare with $z^4$ on the arc and its constant image on the chord. The rest of the actual sector boundary maps into the right half-plane, so its replacement by that chord preserves the winding number. The argument principle then counts exactly one root in the open first quadrant, including multiplicity.

If $f,g$ are holomorphic near the closure of a bounded domain and $|g|<|f|$ on its boundary, then $f$ and $f+g$ have the same number of interior zeros, counted with multiplicity. Apply the argument principle to the zero-free boundary homotopy $f+tg$.

Let $P(z)={\prod }_{\ell }(z-r_{\ell })$ have distinct roots. Choose a disc around $r_j$ with radius $\delta$ smaller than every separation from $r_j$. On its boundary, $|P(z)|\ge \delta {\prod }_{\ell \ne j}(|r_j-r_{\ell }|-\delta )$. If a holomorphic perturbation $H$ satisfies the displayed inequality, Rouché's theorem shows that $P+H$ has exactly one root in that disc, counted with multiplicity. Discs confined to specified half-planes or quadrants give direct geometric root-location proofs without needing a global root-continuation argument.

A locally uniform limit of nonvanishing holomorphic functions on a connected open set is either nonvanishing or identically zero. More generally, isolated zeros persist nearby with their multiplicity.

A nonconstant holomorphic function on a domain maps open sets to open sets. Around any point, isolate its zero relative to the value there and use Rouché's theorem to show that every sufficiently nearby value has a preimage.

If a nonconstant holomorphic function had a local maximum of its modulus, its open image near that point would contain values of larger modulus. Hence no such local maximum exists.

If $f$ is holomorphic near the closed unit disc, $|f|\ge 1$ on its boundary, and $|f(z_0)|<1$ at one interior point, then $f$ maps the open unit disc onto a set containing the open unit disc. Rouché's theorem first shows that $f$ has a zero and then that every $f-w$ with $|w|<1$ has one.

If $g$ is holomorphic and nonvanishing on a punctured disc, thenis the integer winding number of $g(C)$ about zero. If it equals $k$, the logarithmic derivative of $z^{-k}g(z)$ has a removable singularity at zero.