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On $\mathbb{C}\setminus [0,\infty )$, choose the argument in $(0,2\pi )$. Thenis an analytic logarithm. Locally this follows by scaling the convergent series

This extends real exponentiation.

For nonzero complex $z$ and complex $w$, the multivalued power is

An almost complex manifold is a smooth manifold $X$ with a bundle endomorphism $J:TX\to TX$ satisfying $J^2=-I$.

A Fuchsian differential equation has only regular singular points, including possibly the point at infinity.

On a circle of radius $R$, an integrand uniformly of order $R^{-2}$ has contour integral of order $R^{-1}$ by the estimation lemma.

If $F(z)=L+O(1/z)$ uniformly on large circles, then $F(z)/z$ has residue $L$ at infinity in the large-contour sense and its counterclockwise large-circle integral tends to $2\pi iL$.

Two holomorphic functions on a connected domain that agree on a set with an interior accumulation point agree everywhere.

The Dirichlet beta function is $\beta (s)={\sum }_{n\ge 0}(-1{)}^n(2n+1{)}^{-s}$ and is the Dirichlet L-function for the nontrivial character modulo four.

The value of a holomorphic function at the center of a closed disc equals its average around every concentric circle lying in the domain.

The real and imaginary parts are harmonic functions, whose mean-value property explains this average identity.

A nonconstant holomorphic function on a connected domain cannot attain a local maximum of its modulus at an interior point.

A meromorphic function with finitely many specified simple zeros and poles is a rational product up to a nonzero holomorphic factor.

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]

An upper-half-plane self-map is holomorphic and has positive imaginary part in the upper half-plane.

A holomorphic function real on a boundary interval extends across it by conjugate reflection.

Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly.

Suppose $f$ is a holomorphic function on an open set containing the closed disc $|z-a|\le R$, with $R>0$. For any nonnegative integer $n$ and $|w-a|<R$, the counterclockwise boundary contour satisfiesThe case $n=0$ is the Cauchy integral formula. On a smaller closed disc, the denominator stays uniformly away from zero, allowing repeated differentiation under the contour integral. This proves that a holomorphic function has complex derivatives of every order. ^[ref]

At the centre, if $M={\max }_{|z-a|=R}|f(z)|$, taking absolute values gives the Cauchy estimate $|f^{(n)}(a)|\le n!M/R^n$. Boundary size therefore limits interior variation. For $f(z)=z^2$ on the unit disc, $M=1$, and the $n=2$ estimate is exact: $|f^{\prime \prime }(0)|=2$. This is one quantitative reason that holomorphic functions are so rigid.

Every bounded entire function is constant.

A univalent function is a holomorphic injective function.

Contour integration integrates complex functions along oriented curves and evaluates many real integrals through residues.

An isolated singularity at $a$ is a point at which a function is not holomorphic although it is holomorphic throughout some punctured neighbourhood of $a$.