Editing Entire function

{{short description|Function that is holomorphic on the whole complex plane}}
In [[complex analysis]], an '''entire function''', also called an '''integral function,''' is a complex-valued [[Function (mathematics)|function]] that is [[holomorphic function|holomorphic]] on the whole [[complex plane]]. Typical examples of entire functions are [[polynomial]]s and the [[exponential function]], and any finite sums, products and compositions of these, such as the [[trigonometric function]]s [[sine]] and [[cosine]] and their [[hyperbolic function|hyperbolic counterparts]] [[hyperbolic sine|sinh]] and [[hyperbolic cosine|cosh]], as well as [[derivative]]s and [[integral]]s of entire functions such as the [[error function]]. If an entire function <math>f(z)</math> has a 
[[root of a function|root]] at <math>w</math>, then <math>f(z)/(z-w)</math>, taking the limit value at <math>w</math>, 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 [[analytic continuation|continued analytically]] to an entire function.

A '''[[Transcendental function|transcendental]] entire function''' is an entire function that is not a polynomial.

Just as [[Meromorphic function|meromorphic functions]] can be viewed as a generalization of [[Rational function|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's theorem|Mittag-Leffler theorem]] on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the [[Weierstrass factorization theorem|Weierstrass theorem]] on entire functions.

==Properties==
Every entire function <math> f(z) </math> can be represented as a single [[power series]]:
<math display="block">\ f(z) = \sum_{n=0}^\infty a_n z^n\ </math>
that [[convergence (mathematics)|converges]] everywhere in the complex plane, hence [[Compact convergence|uniformly on compact sets]]. The [[radius of convergence]] is infinite, which implies that
<math display="block">\ \lim_{n\to\infty} |a_n|^{\frac{1}{n}} = 0\ </math>
or, equivalently,{{efn|If necessary, the logarithm of zero is taken to be equal to minus infinity.}}
<math display="block">\ \lim_{n\to\infty} \frac{\ln|a_n|}n = -\infty ~.</math>
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 <math>z</math> will be the complex conjugate of the value at <math>z ~.</math> Such functions are sometimes called self-conjugate (the conjugate function, <math>F^*(z),</math> being given by {{nowrap|<math>\bar F(\bar z)</math>).}}{{sfn|Boas|1954|p=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 [[Neighbourhood (mathematics)|neighborhood]] of zero, then we can find the coefficients for <math>n>0</math> from the following derivatives with respect to a real variable <math>r</math>:

<math display="block">\begin{align}
\operatorname\mathcal{Re} \left\{\ a_n\ \right\} &= \frac{1}{n!} \frac{d^n}{dr^n}\ \operatorname\mathcal{Re} \left\{\ f(r)\ \right\} && \quad \mathrm{ at } \quad r = 0 \\
\operatorname\mathcal{Im}\left\{\ a_n\ \right\} &= \frac{1}{n!} \frac{d^n}{dr^n}\ \operatorname\mathcal{Re} \left\{\ f\left( r\ e^{-\frac{i\pi}{2n}} \right)\ \right\} && \quad \mathrm{ at } \quad r = 0
\end{align}</math>

(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.{{efn|
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 <math>i</math> 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 [[zero of a function|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 algebra|unital]] [[associative algebra]] over the [[complex number]]s.

[[Liouville's theorem (complex analysis)|Liouville's theorem]] states that any [[bounded function|bounded]] entire function must be constant.{{efn|
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]]{{efn|
The [[Riemann sphere]] is the whole complex plane augmented with a single point at infinity.
}}
is constant. Thus any non-constant entire function must have a [[mathematical singularity|singularity]] at the complex [[point at infinity]], either a [[pole (complex analysis)|pole]] for a polynomial or an [[essential singularity]] for a [[transcendental function|transcendental]] entire function. Specifically, by the [[Casorati–Weierstrass theorem]], for any transcendental entire function <math>f</math> and any complex <math>w</math> there is a [[sequence]] <math>(z_m)_{m\in\N}</math> such that

:<math>\ \lim_{m\to\infty} |z_m| = \infty, \qquad \text{and} \qquad \lim_{m\to\infty} f(z_m) = w ~.</math>

[[Picard theorem|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 <math>0</math>. One can take a suitable branch of the logarithm of an entire function that never hits <math>0</math>, 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 <math>0</math> 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:

{{math theorem|math_statement= Assume <math>M,</math> <math>R</math> are positive constants and <math>n</math> is a non-negative integer. An entire function <math>f</math> satisfying the inequality <math>|f(z)| \le M |z|^n</math> for all <math>z</math> with <math>|z| \ge R,</math> is necessarily a polynomial, of [[degree of a polynomial|degree]] at most <math>n ~.</math>{{efn|
The converse is also true as for any polynomial <math display="inline">p(z) = \sum_{k=0}^n a_k z^k</math> of degree <math>n</math> the inequality <math display="inline">|p(z)| \le \left(\ \sum_{k=0}^n|a_k|\ \right) |z|^n</math> holds for any <math>|z|\geq 1 ~.</math>
}}
Similarly, an entire function <math>f</math> satisfying the inequality <math>M |z|^n \le |f(z)|</math> for all <math>z</math> with <math>|z| \ge R,</math> is necessarily a polynomial, of degree at least <math>n</math>.}}

==Growth==

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

<math display="block">f(z)=c+\sum_{k=1}^{\infty}\left(\frac{z}{k}\right)^{n_k}</math>

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

=={{anchor|order of an entire function}} Order and type ==
The '''order''' (at infinity) of an entire function <math>f(z)</math> is defined using the [[limit superior]] as:

<math display="block">\rho = \limsup_{r\to\infty}\frac{\ln \left (\ln\| f \|_{\infty, B_r} \right ) }{\ln r},</math>

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

<math display="block">f(z) = O \left (\exp \left (|z|^m \right ) \right ), \quad \text{as } z \to \infty.</math>

The example of <math>f(z) = \exp(2z^2)</math> shows that this does not mean <math>f(z)=O(\exp(|z|^m))</math> if 
<math>f(z)</math> is of order <math>m</math>.

If <math>0<\rho < \infty,</math> one can also define the '''''type''''':

<math display="block">\sigma=\limsup_{r\to\infty}\frac{\ln  \| f\|_{\infty, B_r}} {r^\rho}.</math>

If the order is 1 and the type is <math>\sigma</math>, the function is said to be "of [[exponential type]] <math>\sigma</math>". If it is of order less than 1 it is said to be of exponential type 0.

If <math display="block"> f(z)=\sum_{n=0}^\infty a_n z^n,</math> then the order and type can be found by the formulas
<math display="block">\begin{align}
\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{align}</math>

Let <math>f^{(n)}</math> denote the <math>n</math>-th derivative of <math>f</math>. Then we may restate these formulas in terms of the derivatives at any arbitrary point <math>z_0</math>:

<math display="block">\begin{align}
\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{align}</math>

The type may be infinite, as in the case of the [[reciprocal gamma function]], or zero (see example below under {{slink||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 <math>\rho</math> and <math>\sigma</math> one can construct an example of an entire function of order <math>\rho</math> and type <math>\sigma</math> using:

<math display="block">f(z)=\sum_{n=1}^\infty \left (\frac{e\rho\sigma}{n} \right )^{\frac{n}{\rho}} z^n</math>

====Order 0====
* Non-zero polynomials
*<math>\sum_{n=0}^\infty 2^{-n^2} z^n</math>

====Order 1/4====
<math display="block">f(\sqrt[4]z)</math> 
where <math display="block">f(u)=\cos(u)+\cosh(u)</math>

====Order 1/3====
<math display="block">f(\sqrt[3]z)</math>
where
<math display="block">f(u)=e^u+e^{\omega u}+e^{\omega^2 u} = e^u+2e^{-\frac{u}{2}}\cos \left (\frac{\sqrt 3u}{2} \right ), \quad \text{with } \omega \text{ a complex cube root of 1}.</math>

====Order 1/2====
<math display="block">\cos \left (a\sqrt z \right )</math> with <math>a\neq 0</math> (for which the type is given by <math>\sigma=|a|</math>)

====Order 1====
*<math>\exp(az)</math> with <math>a\neq 0</math> (<math>\sigma=|a|</math>)
*<math>\sin(z)</math>
*<math>\cosh(z)</math>
*the [[Bessel function]]s <math>J_n(z)</math> and spherical Bessel functions <math>j_n(z)</math> for integer values of <math>n</math><ref>See asymptotic expansion in Abramowitz and Stegun, [https://personal.math.ubc.ca/~cbm/aands/page_377.htm p. 377, 9.7.1].</ref>
*the [[reciprocal gamma function]] <math>1/\Gamma(z)</math> (<math>\sigma</math> is infinite)
*<math>\sum_{n=2}^\infty \frac{z^n}{(n\ln n)^n}. \quad (\sigma=0)</math>

====Order 3/2====
* [[Airy function]] <math>Ai(z)</math>

====Order 2====
*<math>\exp(az^2)</math> with <math>a\neq 0</math> (<math>\sigma=|a|</math>)
*The [[Barnes G-function]] (<math>\sigma</math> is infinite).

====Order infinity====
*<math>\exp(\exp(z))</math>

=={{anchor|genus of an entire function}} Genus==
Entire functions of finite order have [[Jacques Hadamard|Hadamard]]'s canonical representation ([[Hadamard factorization theorem]]):

<math display="block">f(z)=z^me^{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),</math>

where <math>z_k</math> are those [[zero of a function|roots]] of <math>f</math> that are not zero (<math>z_k \neq 0</math>), <math>m</math> is the order of the zero of <math>f</math> at <math>z = 0</math> (the case <math>m = 0</math> being taken to mean <math>f(0) \neq 0</math>), <math>P</math> a polynomial (whose degree we shall call <math>q</math>), and <math>p</math> is the smallest non-negative integer such that the series

<math display="block">\sum_{n=1}^\infty\frac{1}{|z_n|^{p+1}}</math>

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

If the order <math>\rho</math> is not an integer, then <math>g = [ \rho ]</math> is the integer part of <math>\rho</math>. If the order is a positive integer, then there are two possibilities: <math>g = \rho-1</math> or <math>g = \rho </math>.

For example, <math>\sin</math>, <math>\cos</math> and <math>\exp</math> are entire functions of genus <math>g = \rho = 1</math>.

==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 integral]]s, 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 [[Paley–Wiener theorem|theorem of Paley and Wiener]], [[Fourier transform]]s of functions (or distributions) with bounded support are entire functions of order <math>1</math> 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 function]]s and [[Parabolic cylinder function]]s arise in this way. The class of entire functions is closed with respect to compositions. This makes it possible to study [[holomorphic dynamics|dynamics of entire functions]].

An entire function of the square root of a complex number is entire if the original function is [[even function|even]], for example <math>\cos(\sqrt{z})</math>.

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, <math>f</math> belongs to this class [[if and only if]] in the Hadamard representation all <math>z_n</math> are real, <math>\rho\leq 1</math>, and 
<math>P(z)=a+bz+cz^2</math>, where <math>b</math> and <math>c</math> are real, and <math>c\leq 0</math>. For example, the sequence of polynomials

<math display="block">\left (1-\frac{(z-d)^2}{n} \right )^n</math>

converges, as <math>n</math> increases, to <math>\exp(-(z-d)^2)</math>. The polynomials

<math display="block"> \frac{1}{2}\left ( \left (1+\frac{iz}{n} \right )^n+ \left (1-\frac{iz}{n} \right )^n \right )</math>

have all real roots, and converge to <math>\cos(z)</math>. The polynomials

<math display="block"> \prod_{m=1}^n \left(1-\frac{z^2}{\left ( \left (m-\frac{1}{2} \right )\pi \right )^2}\right)</math>

also converge to <math>\cos(z)</math>, showing the buildup of the Hadamard product for cosine.

==See also==
* [[Jensen's formula]]
* [[Carlson's theorem]]
* [[Exponential type]]
* [[Paley–Wiener theorem]]
* [[Wiman-Valiron theory]]

==Notes==
{{notelist}}

==References==
{{reflist|25em}}

==Sources==
{{refbegin|colwidth=25em|small=yes}}
* {{cite book
 | first = Ralph P. | last = Boas | author-link = Ralph P. Boas, Jr.
 | year = 1954
 | title = Entire Functions
 | publisher = [[Academic Press]]
 | oclc=847696 | isbn=9780080873138
}}
* {{cite book
 | first = B. Ya. | last = Levin
 | year = 1980 | orig-year=1964
 | title = Distribution of Zeros of Entire Functions
 | publisher = [[American Mathematical Society]]
 | isbn=978-0-8218-4505-9
}}
* {{cite book
 | first = B. Ya. | last = Levin
 | year = 1996
 | title = Lectures on Entire Functions
 | publisher = [[American Mathematical Society]]
 | isbn=978-0-8218-0897-9
}}
{{refend}}

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[[Category:Analytic functions]]
[[Category:Special functions]]