Holomorphic function

A complex-valued function is holomorphic on an open set when it has a complex derivative at every point of that set.

For a holomorphic function nonzero at zero and without boundary zeros, the formula relates its interior zeros, counted by multiplicity, to the circle mean of its logarithmic modulus. Divide out the finite interior zero factors; the remaining logarithmic modulus is harmonic, and each zero contributes $\log (R/|a_j|)$ by the harmonic mean-value formula. Boundary radii follow by limits where appropriate.

Every zero in the smaller disk contributes at least $\log (R/r)$ in Jensen's formula. Bounding the outer circle average by its maximum proves the estimate. With $R=2r$, an exponential $O(r\log r)$ maximum bound gives an $O(r\log r)$ zero count.

A holomorphic primitive of $f$ is a holomorphic function $F$ with $F^{\prime }=f$. A holomorphic $f$ on a domain has such a primitive exactly when its integrals around all closed piecewise smooth curves vanish. Integrating from a fixed starting point then constructs a path-independent primitive.

A function is complex differentiable at $p$ whenIf $f^{\prime }(p)\ne 0$, the real derivative is multiplication by a nonzero complex number, hence a rotation and scaling; this is the local source of angle preservation by holomorphic maps.

For an open set $U\subseteq \mathbb{C}$, the space $\mathcal{O}(U)$ consists of all holomorphic functions on $U$. Its standard topology is locally uniform convergence, equivalently the compact-open topology defined by the seminorms $f\mapsto {\sup }_{z\in K}|f(z)|$ for compact $K\subset U$.

Let $R,\alpha >0$ and ${\mathcal{C}}_{\alpha }=\{(x,t)\in {\mathbb{C}}^2:|x|<R,|t|<\alpha (1-|x|/R)\}$. The displayed norm makes the holomorphic functions with finite norm a Banach space. Finiteness forces $u(x,0)=0$: for fixed $x$, the weight grows like $1/|t|$ near zero. A Cauchy sequence in this norm converges uniformly on every compact subset of ${\mathcal{C}}_{\alpha }$, including at $t=0$. Its limit is holomorphic by locally uniform convergence of holomorphic functions, and the pointwise norm bounds show convergence in the original norm. The shrinking domain compensates for loss of an $x$-derivative in a Cauchy-Kovalevskaya theorem argument.

Write $N=\Vert u{\Vert }_{\alpha }$, $r=|t|$ and $A=\alpha (1-s)>r$. Along the straight integration segment, choose $\sigma (\tau )=s+(A-\tau )/(2\alpha )$. A Cauchy estimate on a disc of radius $R(\sigma -s)$ bounds the integrand by $4\alpha N\tau /[R(A-\tau {)}^2]$. ThereforeThe integral is taken at fixed $x$, and the disc and segment lie inside the shrinking domain. Consequently $u\mapsto {\int }_0^t(i{u}_x+f)dz$ is a contraction mapping when $0<\alpha <R/4$ and the holomorphic source $f$ is bounded on the relevant closed polydisc. The source term has weighted norm at most $\alpha \sup |f|$, so the Banach fixed-point theorem gives a holomorphic solution with zero initial data.

If a nonzero holomorphic function has expansionthen $m$ is the order or multiplicity of its zero at $z_0$.

A simple zero has order one, equivalently $f(z_0)=0$ and $f^{\prime }(z_0)\ne 0$.

An analytic continuation extends a holomorphic function through overlapping connected open sets while preserving its values on their overlap. Continuation along different paths can produce different germs when the domain contains branch points.

Zeta function regularization assigns a finite value to a divergent spectral sum by first forming a convergent complex-power series and then using its analytic continuation. For positive frequencies ${\omega }_n$, continue ${\sum }_n{\omega }_n^{-s}$ to $s=-1$ to define a regulated frequency sum, when that continuation is regular there. This does not turn the divergent positive-term sum into an ordinary convergent sum. It is a useful subtraction convention for the normal-ordering constant of a string and for vacuum energies.

For $r=n+1/2$, $n\ge 0$, the convergent frequency zeta function, expressed through the Riemann zeta function, is ${\sum }_r{r}^{-s}=(2^s-1){\zeta }_R(s)$ when $\mathrm{Re}s>1$. Its analytic continuation at $s=-1$ gives $(1/2-1)(-1/12)=1/24$. A common exponential frequency cutoff gives ${\sum }_{r}r{e}^{-\epsilon r}={\epsilon }^{-2}+1/24+O({\epsilon }^2)$, so the same finite part follows by subtracting the leading divergence. In contrast, integer frequencies have regulated sum $-1/12$.

A meromorphic continuation extends a holomorphic function from a connected open set to a larger connected domain as a meromorphic function, allowing isolated poles. Equality on the initial open set determines the extension uniquely by the identity theorem. Integral formulas whose integrands depend holomorphically on a parameter often supply such an extension after contour deformation or subtraction of singular terms.

As distributions on the real line,The Sokhotski--Plemelj formula relates the two boundary values of a Cauchy transform across its contour and fixes the spectral delta functions generated by retarded and advanced prescriptions.

If a function element can be analytically continued along every path in a domain, continuation along two endpoint-fixed homotopic paths gives the same terminal germ.

For a covering $p:X\to Y$ and base point $y$, lifting loops based at $y$ permutes the fibre $p^{-1}(y)$. The image ofis the monodromy group.

A connected three-sheeted cover has transitive monodromy. If it also has a simple branch point, its monodromy contains a transposition and must be the full symmetric group on three letters. A local triple branch supplies a three-cycle explicitly.

The degree-four rational map $F(z)=z^2+z^{-2}$ has branch values $-2,2,\infty$. Away from their inverse images, the transformationsare commuting deck involutions. They generate a Klein four-group acting transitively on each fibreConsequently the monodromy group is $V_4$, acting by the identity and the three double transpositions.

If a parameter-dependent contour integral has a moving pole, deforming the integration contour continuously so that the pole never crosses it preserves a holomorphic branch. Returning the contour to its original path after crossing a pole adds or subtracts the corresponding residue contribution.

The solutions of $\sinh w=z$ arefor any chosen local branch $\mathrm{Arcsinh}$.

An entire function is a holomorphic function whose domain is the whole complex plane.

A nonzero entire function can be described by its discrete zero locations and multiplicities, a product of Weierstrass elementary factors and a zero-free exponential factor. Conversely, consistent prescribed multiplicities at distinct points escaping every compact subset can be realized by choosing factor orders large enough for infinite product convergence from logarithmic tails. Repeated locations with incompatible exact multiplicities are not admissible data, and a finite accumulation of distinct zeros is excluded by the identity theorem.

An entire function of finite order factors as its canonical zero product times the exponential of a polynomial. For order at most one and nonzero value at zero, it has the form $f(z)=f(0)e^{Bz}{\prod }_j(1-z/a_j)e^{z/a_j}$. Genus-one factors converge when ${\sum }_j|a_j{|}^{-2}<\infty$. A zero-free function of this order is therefore the exponential of an affine polynomial.

If two nonzero entire functions have identical zeros and orders, their quotient extends to a nowhere-zero entire function. Its logarithmic derivative has a primitive on the simply connected plane, producing a holomorphic logarithm $h$ and the displayed relation. Any two choices differ by one constant integer multiple of $2\pi i$. On a multiply connected domain the logarithm can fail to exist globally.

Suppose that for every $z\in \mathbb{C}$, at least one derivative $f^{(n)}(z)$ of an entire function vanishes. The closed sets $E_n=\{z:f^{(n)}(z)=0\}$ cover $\mathbb{C}$. By the Baire category theorem, some $E_N$ has nonempty interior. The identity theorem gives $f^{(N)}=0$ everywhere, so $f$ is a polynomial of degree below $N$.

Every nonconstant entire function takes every complex value with at most one exception.

Near an essential singularity, a holomorphic function takes every complex value, with at most one exception, infinitely often.

If $f=u+iv$ is a holomorphic function on a connected domain andthen the Cauchy-Riemann equations force both components of $f^{\prime }$ to vanish. Hence $f$ is constant.