Entire function

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Entire_function

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function ${\displaystyle f(z)}$ has a root at ${\displaystyle w}$, then ${\displaystyle f(z)/(z-w)}$, taking the limit value at ${\displaystyle w}$, is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function.

A transcendental entire function is an entire function that is not a polynomial.

Just as meromorphic functions can be viewed as a generalization of rational fractions, entire functions can be viewed as a generalization of polynomials. In particular, if for meromorphic functions one can generalize the factorization into simple fractions (the Mittag-Leffler theorem on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the Weierstrass theorem on entire functions.

Properties

Every entire function ${\displaystyle f(z)}$ can be represented as a single power series

$${\displaystyle f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n}$$

that converges everywhere in the complex plane, hence uniformly on compact sets. The radius of convergence is infinite, which implies that

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}|a_n{|}^{\frac{1}{n}}=0}$$

or, equivalently,^[a]

$${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{\ln |a_n|}{n}=-\mathrm{\infty }.}$$

Any power series satisfying this criterion will represent an entire function.

If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the complex conjugate of ${\displaystyle z}$ will be the complex conjugate of the value at ${\displaystyle z.}$ Such functions are sometimes called self-conjugate (the conjugate function, ${\displaystyle F^{\ast }(z),}$ being given by ${\displaystyle \overline{F}(\overline{z})}$).

If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane, up to an imaginary constant. For instance, if the real part is known in a neighborhood of zero, then we can find the coefficients for ${\displaystyle n>0}$ from the following derivatives with respect to a real variable ${\displaystyle r}$:

$${\displaystyle \begin{array}{rlrl}{\mathcal{R}}_{\mathcal{e}}\left\{a_n\right\}& =\frac{1}{n!}\frac{d^n}{d{r}^n}{\mathcal{R}}_{\mathcal{e}}\left\{f(r)\right\}& & \mathrm{a}\mathrm{t}r=0\\ {\mathcal{I}}_{\mathcal{m}}\left\{a_n\right\}& =\frac{1}{n!}\frac{d^n}{d{r}^n}{\mathcal{R}}_{\mathcal{e}}\left\{f\left(r{e}^{-\frac{i\pi }{2n}}\right)\right\}& & \mathrm{a}\mathrm{t}r=0\end{array}}$$

(Likewise, if the imaginary part is known in a neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant. Note however that an entire function is not determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add ${\displaystyle i}$ times any self-conjugate function. If the curve forms a loop, then it is determined by the real part of the function on the loop since the only functions whose real part is zero on the curve are those that are everywhere equal to some imaginary number.

The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes (or "roots").

The entire functions on the complex plane form an integral domain (in fact a Prüfer domain). They also form a commutative unital associative algebra over the complex numbers.

Liouville's theorem states that any bounded entire function must be constant.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere^[d] is constant. Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function ${\displaystyle f}$ and any complex ${\displaystyle w}$ there is a sequence ${\displaystyle (z_m{)}_{m\in \mathbb{N}}}$ such that

${\displaystyle \underset{m\to \mathrm{\infty }}{lim}|z_m|=\mathrm{\infty },\text{and}\underset{m\to \mathrm{\infty }}{lim}f(z_m)=w.}$

Picard's little theorem is a much stronger result: Any non-constant entire function takes on every complex number as value, possibly with a single exception. When an exception exists, it is called a lacunary value of the function. The possibility of a lacunary value is illustrated by the exponential function, which never takes on the value 0 . One can take a suitable branch of the logarithm of an entire function that never hits 0 , so that this will also be an entire function (according to the Weierstrass factorization theorem). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times.

Liouville's theorem is a special case of the following statement:

Theorem — Assume ${\displaystyle M,}$ ${\displaystyle R}$ are positive constants and ${\displaystyle n}$ is a non-negative integer. An entire function ${\displaystyle f}$ satisfying the inequality ${\displaystyle |f(z)|\le M|z{|}^n}$ for all ${\displaystyle z}$ with ${\displaystyle |z|\ge R,}$ is necessarily a polynomial, of degree at most ${\displaystyle n.}$ Similarly, an entire function ${\displaystyle f}$ satisfying the inequality ${\displaystyle M|z{|}^n\le |f(z)|}$ for all ${\displaystyle z}$ with ${\displaystyle |z|\ge R,}$ is necessarily a polynomial, of degree at least ${\displaystyle n}$.

Growth

Entire functions may grow as fast as any increasing function: for any increasing function ${\displaystyle g:[0,\mathrm{\infty })\to [0,\mathrm{\infty })}$ there exists an entire function ${\displaystyle f}$ such that ${\displaystyle f(x)>g(|x|)}$ for all real ${\displaystyle x}$. Such a function ${\displaystyle f}$ may be easily found of the form:

$${\displaystyle f(z)=c+\sum _{k=1}^{\mathrm{\infty }}{\left(\frac{z}{k}\right)}^{n_k}}$$

for a constant ${\displaystyle c}$ and a strictly increasing sequence of positive integers ${\displaystyle n_k}$. Any such sequence defines an entire function ${\displaystyle f(z)}$, and if the powers are chosen appropriately we may satisfy the inequality ${\displaystyle f(x)>g(|x|)}$ for all real ${\displaystyle x}$. (For instance, it certainly holds if one chooses ${\displaystyle c:=g(2)}$ and, for any integer ${\displaystyle k\ge 1}$ one chooses an even exponent ${\displaystyle n_k}$ such that ${\displaystyle {\left(\frac{k+1}{k}\right)}^{n_k}\ge g(k+2)}$).

Order and type

The order (at infinity) of an entire function ${\displaystyle f(z)}$ is defined using the limit superior as:

$${\displaystyle \rho =\underset{r\to \mathrm{\infty }}{lim sup}\frac{\ln \left(\ln \Vert f{\Vert }_{\mathrm{\infty },B_r}\right)}{\ln r},}$$

where ${\displaystyle B_r}$ is the disk of radius ${\displaystyle r}$ and ${\displaystyle \Vert f{\Vert }_{\mathrm{\infty },B_r}}$ denotes the supremum norm of ${\displaystyle f(z)}$ on ${\displaystyle B_r}$. The order is a non-negative real number or infinity (except when ${\displaystyle f(z)=0}$ for all ${\displaystyle z}$. In other words, the order of ${\displaystyle f(z)}$ is the infimum of all ${\displaystyle m}$ such that:

$${\displaystyle f(z)=O\left(\exp \left(|z{|}^m\right)\right),\text{as }z\to \mathrm{\infty }.}$$

The example of ${\displaystyle f(z)=\exp (2z^2)}$ shows that this does not mean ${\displaystyle f(z)=O(\exp (|z{|}^m))}$ if ${\displaystyle f(z)}$ is of order ${\displaystyle m}$.

If ${\displaystyle 0<\rho <\mathrm{\infty },}$ one can also define the type:

$${\displaystyle \sigma =\underset{r\to \mathrm{\infty }}{lim sup}\frac{\ln \Vert f{\Vert }_{\mathrm{\infty },B_r}}{r^{\rho }}.}$$

If the order is 1 and the type is ${\displaystyle \sigma }$, the function is said to be "of exponential type ${\displaystyle \sigma }$". If it is of order less than 1 it is said to be of exponential type 0.

If

$${\displaystyle f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n,}$$

then the order and type can be found by the formulas

$${\displaystyle \begin{array}{rl}\rho & =\underset{n\to \mathrm{\infty }}{lim sup}\frac{n\ln n}{-\ln |a_n|}\\ (e\rho \sigma {)}^{\frac{1}{\rho }}& =\underset{n\to \mathrm{\infty }}{lim sup}{n}^{\frac{1}{\rho }}|a_n{|}^{\frac{1}{n}}\end{array}}$$

Let ${\displaystyle f^{(n)}}$ denote the ${\displaystyle n}$-th derivative of ${\displaystyle f}$, then we may restate these formulas in terms of the derivatives at any arbitrary point ${\displaystyle z_0}$:

$${\displaystyle \begin{array}{rl}\rho & =\underset{n\to \mathrm{\infty }}{lim sup}\frac{n\ln n}{n\ln n-\ln |f^{(n)}(z_0)|}={\left(1-\underset{n\to \mathrm{\infty }}{lim sup}\frac{\ln |f^{(n)}(z_0)|}{n\ln n}\right)}^{-1}\\ (\rho \sigma {)}^{\frac{1}{\rho }}& =e^{1-\frac{1}{\rho }}\underset{n\to \mathrm{\infty }}{lim sup}\frac{|f^{(n)}(z_0){|}^{\frac{1}{n}}}{n^{1-\frac{1}{\rho }}}\end{array}}$$

The type may be infinite, as in the case of the reciprocal gamma function, or zero (see example below under § Order 1).

Another way to find out the order and type is Matsaev's theorem.

Examples

Here are some examples of functions of various orders:

Order ρ

For arbitrary positive numbers ${\displaystyle \rho }$ and ${\displaystyle \sigma }$ one can construct an example of an entire function of order ${\displaystyle \rho }$ and type ${\displaystyle \sigma }$ using:

$${\displaystyle f(z)=\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{e\rho \sigma }{n}\right)}^{\frac{n}{\rho }}{z}^n}$$

Order 0

• Non-zero polynomials
• ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{2}^{-n^2}{z}^n}$

Order 1/4

$${\displaystyle f(\sqrt[4]{z})}$$

where

$${\displaystyle f(u)=\cos (u)+\cosh (u)}$$

Order 1/3

$${\displaystyle f(\sqrt[3]{z})}$$

where

$${\displaystyle f(u)=e^u+e^{\omega u}+e^{{\omega }^{2}u}=e^u+2e^{-\frac{u}{2}}\cos \left(\frac{\sqrt{3}u}{2}\right),\text{with }\omega \text{ a complex cube root of 1}.}$$

Order 1/2

$${\displaystyle \cos \left(a\sqrt{z}\right)}$$

with ${\displaystyle a\ne 0}$ (for which the type is given by ${\displaystyle \sigma =|a|}$)

Order 1

• ${\displaystyle \exp (az)}$ with ${\displaystyle a\ne 0}$ (${\displaystyle \sigma =|a|}$)
• ${\displaystyle \sin (z)}$
• ${\displaystyle \cosh (z)}$
• the Bessel function ${\displaystyle J_0(z)}$
• the reciprocal gamma function ${\displaystyle 1/\mathrm{\Gamma }(z)}$ (${\displaystyle \sigma }$ is infinite)
• ${\displaystyle \sum _{n=2}^{\mathrm{\infty }}\frac{z^n}{(n\ln n{)}^n}.(\sigma =0)}$

Order 3/2

• Airy function ${\displaystyle Ai(z)}$

Order 2

• ${\displaystyle \exp (a{z}^2)}$ with ${\displaystyle a\ne 0}$ (${\displaystyle \sigma =|a|}$)
• The Barnes G-function (${\displaystyle \sigma }$ is infinite).

Order infinity

• ${\displaystyle \exp (\exp (z))}$

Genus

Entire functions of finite order have Hadamard's canonical representation (Hadamard factorization theorem):

$${\displaystyle f(z)=z^m{e}^{P(z)}\prod _{n=1}^{\mathrm{\infty }}\left(1-\frac{z}{z_n}\right)\exp \left(\frac{z}{z_n}+\cdots +\frac{1}{p}{\left(\frac{z}{z_n}\right)}^p\right),}$$

where ${\displaystyle z_k}$ are those roots of ${\displaystyle f}$ that are not zero (${\displaystyle z_k\ne 0}$), ${\displaystyle m}$ is the order of the zero of ${\displaystyle f}$ at ${\displaystyle z=0}$ (the case ${\displaystyle m=0}$ being taken to mean ${\displaystyle f(0)\ne 0}$), ${\displaystyle P}$ a polynomial (whose degree we shall call ${\displaystyle q}$), and ${\displaystyle p}$ is the smallest non-negative integer such that the series

$${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{|z_n{|}^{p+1}}}$$

converges. The non-negative integer ${\displaystyle g=max\{p,q\}}$ is called the genus of the entire function ${\displaystyle f}$.

If the order ${\displaystyle \rho }$ is not an integer, then ${\displaystyle g=[\rho ]}$ is the integer part of ${\displaystyle \rho }$. If the order is a positive integer, then there are two possibilities: ${\displaystyle g=\rho -1}$ or ${\displaystyle g=\rho }$.

For example, ${\displaystyle \sin }$, ${\displaystyle \cos }$ and ${\displaystyle \exp }$ are entire functions of genus ${\displaystyle g=\rho =1}$.

Other examples

According to J. E. Littlewood, the Weierstrass sigma function is a 'typical' entire function. This statement can be made precise in the theory of random entire functions: the asymptotic behavior of almost all entire functions is similar to that of the sigma function. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and the error function are special cases of the Mittag-Leffler function. According to the fundamental theorem of Paley and Wiener, Fourier transforms of functions (or distributions) with bounded support are entire functions of order ${\displaystyle 1}$ and finite type.

Other examples are solutions of linear differential equations with polynomial coefficients. If the coefficient at the highest derivative is constant, then all solutions of such equations are entire functions. For example, the exponential function, sine, cosine, Airy functions and Parabolic cylinder functions arise in this way. The class of entire functions is closed with respect to compositions. This makes it possible to study dynamics of entire functions.

An entire function of the square root of a complex number is entire if the original function is even, for example ${\displaystyle \cos (\sqrt{z})}$.

If a sequence of polynomials all of whose roots are real converges in a neighborhood of the origin to a limit which is not identically equal to zero, then this limit is an entire function. Such entire functions form the Laguerre–Pólya class, which can also be characterized in terms of the Hadamard product, namely, ${\displaystyle f}$ belongs to this class if and only if in the Hadamard representation all ${\displaystyle z_n}$ are real, ${\displaystyle \rho \le 1}$, and ${\displaystyle P(z)=a+bz+c{z}^2}$, where ${\displaystyle b}$ and ${\displaystyle c}$ are real, and ${\displaystyle c\le 0}$. For example, the sequence of polynomials

$${\displaystyle {\left(1-\frac{(z-d{)}^2}{n}\right)}^n}$$

converges, as ${\displaystyle n}$ increases, to ${\displaystyle \exp (-(z-d{)}^2)}$. The polynomials

$${\displaystyle \frac{1}{2}\left({\left(1+\frac{iz}{n}\right)}^n+{\left(1-\frac{iz}{n}\right)}^n\right)}$$

have all real roots, and converge to ${\displaystyle \cos (z)}$. The polynomials

$${\displaystyle \prod _{m=1}^n\left(1-\frac{z^2}{{\left(\left(m-\frac{1}{2}\right)\pi \right)}^2}\right)}$$

also converge to ${\displaystyle \cos (z)}$, showing the buildup of the Hadamard product for cosine.

Holomorphic function

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Holomorphic_function

A rectangular grid (top) and its image under a conformal map f (bottom).

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cⁿ. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central objects of study in complex analysis.

Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.

Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z₀" means not just differentiable at z₀, but differentiable everywhere within some neighbourhood of z₀ in the complex plane.

Definition

The function f(z) = z̅ is not complex differentiable at zero, because as shown above, the value of (f(z) − f(0)) / (z − 0) varies depending on the direction from which zero is approached. Along the real axis, f equals the function g(z) = z and the limit is 1, while along the imaginary axis, f equals h(z) = −z and the limit is −1. Other directions yield yet other limits.

Given a complex-valued function f of a single complex variable, the derivative of f at a point z₀ in its domain is defined as the limit

${\displaystyle f^{\prime }(z_0)=\underset{z\to z_0}{lim}\frac{f(z)-f(z_0)}{z-z_0}.}$

This is the same definition as for the derivative of a real function, except that all quantities are complex. In particular, the limit is taken as the complex number z tends to z₀, and this means that the same value is obtained for any sequence of complex values for z that tends to z₀. If the limit exists, f is said to be complex differentiable at z₀. This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.

A function is holomorphic on an open set U if it is complex differentiable at every point of U. A function f is holomorphic at a point z₀ if it is holomorphic on some neighbourhood of z₀. A function is holomorphic on some non-open set A if it is holomorphic at every point of A.

A function may be complex differentiable at a point but not holomorphic at this point. For example, the function ${\displaystyle f(z)=|z{|}^2=z\overline{z}}$ is complex differentiable at 0, but not complex differentiable elsewhere (see the Cauchy–Riemann equations, below). So, it is not holomorphic at 0.

The relationship between real differentiability and complex differentiability is the following: If a complex function f(x + i y) = u(x, y) + i v(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy–Riemann equations:

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }x}=\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\text{and}\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{\mathrm{\partial }v}{\mathrm{\partial }x}}$

or, equivalently, the Wirtinger derivative of f with respect to ${\displaystyle \overline{z},}$ the complex conjugate of ${\displaystyle z,}$ is zero:

${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }\overline{z}}=0,}$

which is to say that, roughly, f is functionally independent from ${\displaystyle \overline{z},}$ the complex conjugate of z.

If continuity is not given, the converse is not necessarily true. A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then f is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if f is continuous, u and v have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then f is holomorphic.

Terminology

The term holomorphic was introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Augustin-Louis Cauchy's students, and derives from the Greek ὅλος (hólos) meaning "whole", and μορφή (morphḗ) meaning "form" or "appearance" or "type", in contrast to the term meromorphic derived from μέρος (méros) meaning "part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane. Cauchy had instead used the term synectic.

Today, the term "holomorphic function" is sometimes preferred to "analytic function". An important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.

Properties

Because complex differentiation is linear and obeys the product, quotient, and chain rules, the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero. That is, if functions f and g are holomorphic in a domain U, then so are f + g, f − g, f g, and f ∘ g. Furthermore, f / g is holomorphic if g has no zeros in U, or is meromorphic otherwise.

If one identifies C with the real plane R², then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy–Riemann equations, a set of two partial differential equations.

Every holomorphic function can be separated into its real and imaginary parts f(x + i y) = u(x, y) + i v(x, y), and each of these is a harmonic function on R² (each satisfies Laplace's equation ∇² u = ∇² v = 0), with v the harmonic conjugate of u. Conversely, every harmonic function u(x, y) on a simply connected domain Ω ⊂ R² is the real part of a holomorphic function: If v is the harmonic conjugate of u, unique up to a constant, then f(x + i y) = u(x, y) + i v(x, y) is holomorphic.

Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes:

${\displaystyle {\oint }_{\gamma }f(z)dz=0.}$

Here γ is a rectifiable path in a simply connected complex domain U ⊂ C whose start point is equal to its end point, and f : U → C is a holomorphic function.

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary. Furthermore: Suppose U ⊂ C is a complex domain, f : U → C is a holomorphic function and the closed disk D = {z : |z − z₀| ≤ r} is completely contained in U. Let γ be the circle forming the boundary of D. Then for every a in the interior of D:

${\displaystyle f(a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{z-a}dz}$

where the contour integral is taken counter-clockwise.

The derivative f′(a) can be written as a contour integral using Cauchy's differentiation formula:

${\displaystyle f^{\prime }(a)=\frac{1}{2\pi i}{\oint }_{\gamma }\frac{f(z)}{(z-a{)}^2}dz,}$

for any simple loop positively winding once around a, and

${\displaystyle f^{\prime }(a)=\underset{\gamma \to a}{lim}\frac{i}{2\mathcal{A}(\gamma )}{\oint }_{\gamma }f(z)d\overline{z},}$

for infinitesimal positive loops γ around a.

In regions where the first derivative is not zero, holomorphic functions are conformal: they preserve angles and the shape (but not size) of small figures.

Every holomorphic function is analytic. That is, a holomorphic function f has derivatives of every order at each point a in its domain, and it coincides with its own Taylor series at a in a neighbourhood of a. In fact, f coincides with its Taylor series at a in any disk centred at that point and lying within the domain of the function.

From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space. Additionally, the set of holomorphic functions in an open set U is an integral domain if and only if the open set U is connected.  In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

From a geometric perspective, a function f is holomorphic at z₀ if and only if its exterior derivative df in a neighbourhood U of z₀ is equal to f′(z) dz for some continuous function f′. It follows from

${\displaystyle 0=d^{2}f=d(f^{\mathrm{\prime }}dz)=d{f}^{\mathrm{\prime }}\wedge dz}$

that df′ is also proportional to dz, implying that the derivative f′ is itself holomorphic and thus that f is infinitely differentiable. Similarly, d(f dz) = f′ dz ∧ dz = 0 implies that any function f that is holomorphic on the simply connected region U is also integrable on U.

(For a path γ from z₀ to z lying entirely in U, define $F_{\gamma }(z)=F_0+{\int }_{\gamma }fdz;$ in light of the Jordan curve theorem and the generalized Stokes' theorem, F_γ(z) is independent of the particular choice of path γ, and thus F(z) is a well-defined function on U having F(z₀) = F₀ and dF = f dz.)

Examples

All polynomial functions in z with complex coefficients are entire functions (holomorphic in the whole complex plane C), and so are the exponential function exp z and the trigonometric functions $\cos z=\frac{1}{2}(\exp (iz)+\exp (-iz))$ and $\sin z=-\frac{1}{2}i(\exp (iz)-\exp (-iz))$ (cf. Euler's formula). The principal branch of the complex logarithm function log z is holomorphic on the domain C ∖ {z ∈ R : z ≤ 0}. The square root function can be defined as $\sqrt{z}=\exp (\frac{1}{2}\log z)$ and is therefore holomorphic wherever the logarithm log z is. The reciprocal function 1 / z is holomorphic on C ∖ {0}. (The reciprocal function, and any other rational function, is meromorphic on C.)

As a consequence of the Cauchy–Riemann equations, any real-valued holomorphic function must be constant. Therefore, the absolute value |z |, the argument arg (z), the real part Re (z) and the imaginary part Im (z) are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate ${\displaystyle \overline{z}.}$ (The complex conjugate is antiholomorphic.)

Several variables

The definition of a holomorphic function generalizes to several complex variables in a straightforward way. A function ${\displaystyle f:(z_1,z_2,\dots ,z_n)\mapsto f(z_1,z_2,\dots ,z_n)}$ in n complex variables is analytic at a point p if there exists a neighbourhood of p in which f is equal to a convergent power series in n complex variables; the function f is holomorphic in an open subset U of Cⁿ if it is analytic at each point in U. Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function f, this is equivalent to f being holomorphic in each variable separately (meaning that if any n − 1 coordinates are fixed, then the restriction of f is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that the continuity assumption is unnecessary: f is holomorphic if and only if it is holomorphic in each variable separately.

More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.

Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are logarithmically-convex Reinhardt domains, the simplest example of which is a polydisk. However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a domain of holomorphy.

A complex differential (p, 0)-form α is holomorphic if and only if its antiholomorphic Dolbeault derivative is zero: ∂α = 0.

Extension to functional analysis

Main article: infinite-dimensional holomorphy

The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gateaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.