# Entire function

Last updated

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 f(z) has a root at w, then f(z)/(z−w), taking the limit value at w, is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.

## Contents

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

## Properties

Every entire function f(z) can be represented as a power series

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

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

${\displaystyle \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}=0}$

or

${\displaystyle \lim _{n\to \infty }{\frac {\ln |a_{n}|}{n}}=-\infty .}$

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 z will be the complex conjugate of the value at z. Such functions are sometimes called self-conjugate (the conjugate function, ${\displaystyle F^{*}(z)}$, being given by ${\displaystyle {\bar {F}}({\bar {z}})}$). [1]

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 n > 0 from the following derivatives with respect to a real variable r:

{\displaystyle {\begin{aligned}\operatorname {Re} a_{n}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\operatorname {Re} f(r)&&{\text{at }}r=0\\\operatorname {Im} a_{n}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\operatorname {Re} f\left(re^{-{\frac {i\pi }{2n}}}\right)&&{\text{at }}r=0\end{aligned}}}

(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. (For instance, if the real part is known on part of the unit circle, then it is known on the whole unit circle by analytic extension, and then the coefficients of the infinite series are determined from the coefficients of the Fourier series for the real part on the unit circle.) 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 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. Liouville's theorem may be used to elegantly prove the fundamental theorem of algebra.

As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere (complex plane and the point at infinity) 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 f and any complex w there is a sequence ${\displaystyle (z_{m})_{m\in \mathbb {N} }}$ such that

${\displaystyle \lim _{m\to \infty }|z_{m}|=\infty ,\qquad {\text{and}}\qquad \lim _{m\to \infty }f(z_{m})=w.}$

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 M, R are positive constants and n is a non-negative integer. An entire function f satisfying the inequality ${\displaystyle |f(z)|\leq M|z|^{n}}$ for all z with ${\displaystyle |z|\geq R,}$ is necessarily a polynomial, of degree at most n. [2] Similarly, an entire function f satisfying the inequality ${\displaystyle M|z|^{n}\leq |f(z)|}$ for all z with ${\displaystyle |z|\geq R,}$ is necessarily a polynomial, of degree at least n.

## Growth

Entire functions may grow as fast as any increasing function: for any increasing function g: [0,∞) → [0,∞) there exists an entire function f such that f(x) > g(|x|) for all real x. Such a function f may be easily found of the form:

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

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

## Order and type

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

${\displaystyle \rho =\limsup _{r\to \infty }{\frac {\ln \left(\ln \|f\|_{\infty ,B_{r}}\right)}{\ln r}},}$

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

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

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

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

${\displaystyle \sigma =\limsup _{r\to \infty }{\frac {\ln \|f\|_{\infty ,B_{r}}}{r^{\rho }}}.}$

If the order is 1 and the type is σ, the function is said to be "of exponential type σ". If it is of order less than 1 it is said to be of exponential type 0.

If

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

then the order and type can be found by the formulas

{\displaystyle {\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{-\ln |a_{n}|}}\\[6pt](e\rho \sigma )^{\frac {1}{\rho }}&=\limsup _{n\to \infty }n^{\frac {1}{\rho }}|a_{n}|^{\frac {1}{n}}\end{aligned}}}

Let ${\displaystyle f^{(n)}}$ denote the nth derivative of f, then we may restate these formulas in terms of the derivatives at any arbitrary point z0:

{\displaystyle {\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{n\ln n-\ln |f^{(n)}(z_{0})|}}=\left(1-\limsup _{n\to \infty }{\frac {\ln |f^{(n)}(z_{0})|}{n\ln n}}\right)^{-1}\\[6pt](\rho \sigma )^{\frac {1}{\rho }}&=e^{1-{\frac {1}{\rho }}}\limsup _{n\to \infty }{\frac {|f^{(n)}(z_{0})|^{\frac {1}{n}}}{n^{1-{\frac {1}{\rho }}}}}\end{aligned}}}

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

### Examples

Here are some examples of functions of various orders:

#### Order ρ

For arbitrary positive numbers ρ and σ one can construct an example of an entire function of order ρ and type σ using:

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

#### Order 0

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

#### 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

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

#### Order 1/2

${\displaystyle \cos \left(a{\sqrt {z}}\right)}$ with a ≠ 0 (for which the type is given by σ = |a|)

#### Order 2

• exp(az2) with a ≠ 0 (σ = |a|)

• exp(exp(z))

## Genus

Entire functions of finite order have Hadamard's canonical representation:

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

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

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

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

If the order ρ is not an integer, then ${\displaystyle g=\lbrack \rho \rbrack }$ 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, ${\displaystyle \sin ,\cos }$ and ${\displaystyle \exp }$ are entire functions of genus 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 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, f belongs to this class if and only if in the Hadamard representation all zn are real, p ≤ 1, and P(z) = a + bz + cz2, where b and c are real, and c ≤ 0. For example, the sequence of polynomials

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

converges, as n increases, to exp(−(zd)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 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 cos(z), showing the buildup of the Hadamard product for cosine.

## Notes

1. For example, ( Boas 1954 , p. 1)
2. The converse is also true as for any polynomial ${\displaystyle p(z)=\sum _{k=0}^{n}a_{k}z^{k}}$ of degree n the inequality ${\displaystyle |p(z)|\leq \left(\sum _{k=0}^{n}|a_{k}|\right)|z|^{n}}$ holds for any |z| ≥ 1.

## Related Research Articles

In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.

In mathematics, the exponential function is the function where e = 2.71828... is Euler's number.

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For any positive integer n,

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, logex, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

In mathematics, the ratio test is a test for the convergence of a series

In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense, and why the characterizations are independent of and equivalent to each other. As a special case of these considerations, it will be demonstrated that the three most common definitions given for the mathematical constant e are equivalent to each other.

In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a product involving its zeroes. The theorem may be viewed as an extension of the fundamental theorem of algebra, which asserts that every polynomial may be factored into linear factors, one for each root.

In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by Riemann (1859) for the Riemann zeta function. Such explicit formulae have been applied also to questions on bounding the discriminant of an algebraic number field, and the conductor of a number field.

A ratio distribution is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two random variables X and Y, the distribution of the random variable Z that is formed as the ratio Z = X/Y is a ratio distribution.

In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In statistics, the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis, and other applications of multivariate statistical analysis on metric spaces. It is named after the Swedish forestry statistician Bertil Matérn. It is commonly used to define the statistical covariance between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the Matérn covariance is also isotropic.

In the field of mathematical analysis, a general Dirichlet series is an infinite series that takes the form of

In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region if and only if is analytic on and has a harmonic majorant on where . Being the ratio of two bounded analytic functions is a sufficient condition for a function to be of bounded type, and if is simply connected the condition is also necessary.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

The Laguerre–Pólya class is the class of entire functions consisting of those functions which are locally the limit of a series of polynomials whose roots are all real. Any function of Laguerre–Pólya class is also of Pólya class.

## References

• Boas, Ralph P. (1954). Entire Functions. Academic Press. ISBN   9780080873138. OCLC   847696.
• Levin, B. Ya. (1980) [1964]. Distribution of zeros of entire functions. Amer. Math. Soc. ISBN   978-0-8218-4505-9.
• Levin, B. Ya. (1996). Lectures on entire functions. Amer. Math. Soc. ISBN   978-0-8218-0897-9.