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| In [[complex analysis]], an '''entire function''', also called an '''integral function,''' is a complex-valued [[Function (mathematics)|function]] that is [[holomorphic function|holomorphic]] over the whole [[complex plane]]. Typical examples of entire functions are the [[polynomial]]s and the [[exponential function]], and any 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 ''f''(''z'') has a [[root of a function|root]] at ''w'', then ''f''(''z'')/(''z−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 [[analytic continuation|continued analytically]] to an entire function.
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| A '''[[Transcendental function|transcendental]] entire function''' is an entire function that is not a polynomial.
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| ==Properties==
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| Every entire function ''f''(''z'') can be represented as a [[power series]]
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| :<math>f(z)=\sum_{n=0}^\infty a_nz^n</math>
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| that converges everywhere in the complex plane, hence [[Compact convergence|uniformly on compact sets]]. The [[radius of convergence]] is infinite, which implies that
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| :<math>\lim_{n\to\infty} \left|a_n\right|^{\frac{1}{n}}=0</math>
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| or
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| :<math>\lim_{n\to\infty}\frac{\ln|a_n|}n=-\infty.</math>
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| Any power series satisfying this criterion will represent an entire function.
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| The [[Weierstrass factorization theorem]] asserts that any entire function can be represented by a product involving its [[zero (complex analysis)|zeroes]] (or "roots").
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| 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.
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| [[Liouville's theorem (complex analysis)|Liouville's theorem]] states that any [[bounded function|bounded]] entire function must be constant. Liouville's theorem may be used to elegantly prove the [[fundamental theorem of algebra]].
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| 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 [[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 ''f'' and any complex ''w'' there is a [[sequence]] {{nowrap|(''z<sub>m</sub>'')<sub>''m''∈'''N'''</sub>}} with <math>\lim_{m\to\infty} |z_m| = \infty</math> and <math>\lim_{m\to\infty} f(z_m) = w\ </math>.
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| [[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. The latter exception is illustrated by the [[exponential function]], which never takes on the value 0. One can take a logarithm of an entire function that never hits 0, and this will also be an entire function (according to the [[Weierstrass factorization theorem]]). The logarithm hits every complex number except possibly one, which implies that the first function will hit any value other than 0 an infinite number of times. Similarly, an entire function that does not hit a particular value will hit every other value an infinite number of times.
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| Liouville's theorem is a special case of the following statement:
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| <blockquote>'''Theorem:''' Assume ''M, R'' are positive constants and that ''n'' is a non-negative integer. An entire function ''f'' satisfying the inequality <math>|f(z)| \le M |z|^n</math> for all ''z'' with <math>|z| \ge R</math>, is necessarily a polynomial, of [[degree of a polynomial|degree]] at most ''n''.<ref>The converse is also true as for any polynomial <math>\textstyle p(z) = \sum _{k=0}^na_k z^k</math> of degree ''n'' the inequality <math>\textstyle |p(z)| \le \left(\sum_{k=0}^n|a_k|\right) |z|^n</math> holds for any {{nowrap begin}}|''z''| ≥ 1{{nowrap end}}.</ref> Similarly, an entire function ''f'' satisfying the inequality <math>M |z|^n \le |f(z)|</math> for all ''z'' with <math>|z| \ge R</math>, is necessarily a polynomial, of degree at least ''n''.</blockquote>
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| If (and only if) the coefficients of the power series are all real then the function (obviously) 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, <math>F^*(z)</math>, being given by <math>\bar F(\bar z)).</math>
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| ==Growth==
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| Entire functions may grow as fast as any increasing function: for any increasing function ''g'' : [0,+∞) → [0,+∞) there exists an entire function ''f(z)'' such that ''f''(''x'') > ''g''( |''x''| ) for all real ''x''. Such a function ''f'' may be easily found of the form:
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| :<math>f(z)=c+\sum_{k=1}^{\infty}\left(\frac{z}{k}\right)^{n_k}</math>
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| for a constant ''c'' and a strictly increasing sequence of positive integers ''n<sub>k</sub>''. 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 ''k ≥ 1'', <math>n_k: = 2 \big\lceil k \ln g(k+2) \big\rceil \,</math> although this gives powers that may be about twice as high as needed.)
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| ==<span id="order of an entire function"></span> Order and type ==
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| The '''order''' (at infinity) of an entire function ''f(z)'' is defined using the [[limit superior]] as: | |
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| :<math>\rho=\limsup_{r\rightarrow\infty}\frac{\ln(\ln\Vert f \Vert_{\infty, B_r} )}{\ln\, r},</math>
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| where ''B<sub>r</sub>'' is the disk of radius ''r'' and <math> \Vert f \Vert_{\infty,\,B_r}</math> denotes the [[supremum norm]] of ''f(z)'' on ''B<sub>r</sub>''. The order is a non-negative real number or infinity (except if ''f(z)''=0 for all ''z''). The order of ''f''(''z'') is the [[infimum]] of all ''m'' such that ''f(z)'' = O(exp(|''z''|<sup>''m''</sup>)) as ''z'' → ∞. (A function like <math>\exp(2z^2)</math> shows that this does not mean ''f(z)'' = O(exp(|''z''|<sup>''m''</sup>)) if ''f(z)'' is of order ''m''.)
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| If 0<ρ<∞, one can also define the '''''type''''':
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| :<math>\sigma=\limsup_{r\rightarrow\infty}\frac{\ln \Vert f\Vert_{\infty,B_r}} {r^\rho}.</math>
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| If
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| :<math> f(z)=\sum_{n=0}^\infty a_n z^n,</math>
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| then the order and type can be found by the formulas
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| :<math> \rho=\limsup_{n\to\infty}\frac{n\ln n}{-\ln|a_n|},</math> | |
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| :<math> (e\rho\sigma)^{1/\rho} = \limsup_{n\to\infty} n^{1/\rho} |a_n|^{1/n}.</math>
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| If we denote the ''n''<sup>th</sup> derivative of a function ''f''() by ''f''<sup>(''n'')</sup>(), then we may restate these formulas in terms of the derivatives at any arbitrary point ''z''<sub>0</sub>:
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| :<math> \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},</math>
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| :<math>(\rho\sigma)^{1/\rho} =e^{1-1/\rho} \limsup_{n\to\infty}\frac{|f^{(n)}(z_0)|^{1/n}}{n^{1-1/\rho}}.</math> | |
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| The type may be infinite, as in the case of the [[reciprocal gamma function]], or zero (see example below under [[#Order 1]]).
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| ===Examples===
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| Here are some examples of functions of various orders:
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| ====Order ρ====
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| For arbitrary positive numbers ρ and σ one can construct an example of an entire function of order ρ and type σ using:
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| :<math>f(z)=\sum_{n=1}^\infty (e\rho\sigma/n)^{n/\rho} z^n</math>
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| ====Order 0====
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| *Polynomials (other than 0)
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| *<math>\sum_{n=0}^\infty 2^{-n^2} z^n</math>
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| ====Order 1/4====
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| *<math>f(\sqrt[4]z)\text{ where }f(u)=\cos(u)+\cosh(u)</math>
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| ====Order 1/3====
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| *<math>f(\sqrt[3]z)\text{ where }f(u)=e^u+e^{\omega u}+e^{\omega^2 u}=e^u+2e^{-u/2}\cos(\sqrt 3u/2),\text{with }\omega\text{ a complex cube root of 1}</math>
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| ====Order 1/2====
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| *<math>\cos(a\sqrt z)</math> with ''a'' ≠ 0 (for which the type is given by σ = |''a''|)
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| ====Order 1====
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| *exp(''az'') with ''a'' ≠ 0 (σ = |''a''|)
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| *sin(''z'')
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| *cosh(''z'')
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| *[[Bessel function]] ''J''<sub>0</sub>(''z'')
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| *the [[reciprocal gamma function]] 1/Γ(''z'') (σ is infinite)
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| *<math>\sum_{n=2}^\infty z^n/(n\ln n)^n.\ (\sigma=0)</math>
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| ====Order 3/2====
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| * [[Airy function]] Ai(''z'')
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| ====Order 2====
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| *exp(−''az''<sup>2</sup>) with ''a'' ≠ 0 (σ = |''a''|)
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| ====Order infinity====
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| *exp(−''e<sup>z</sup>'')
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| ==<span id="genus of an entire function"></span> Genus of an entire function ==
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| Entire functions of finite order have [[Jacques Hadamard|Hadamard]]'s canonical representation:
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| :<math>f(z)=z^me^{P(z)}\prod_{n=1}^\infty\left(1-\frac{z}{z_n}\right)\exp\left(\frac{z}{z_n}+\ldots+\frac{1}{p}\left(\frac{z}{z_n}\right)^p\right),</math>
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| where ''z<sub>k</sub>'' are the non-zero [[zero (complex analysis)|roots]] of ''f'', ''P'' a polynomial (whose degree we shall call ''q''), and ''p'' is the smallest non-negative integer such that the series
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| :<math>\sum_{n=1}^\infty\frac{1}{|z_n|^{p+1}}</math>
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| converges. The non-negative integer ''g'' = max{''p'', ''q''} is called the genus of the entire function ''f''.
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| If the order ρ is not an integer, then ''g'' = [ρ] is the integer part of ρ. If the order is a positive integer, then there are two possibilities: ''g'' = [ρ] or ''g'' =[ρ]+1.
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| For example, ''sin'', ''cos'' and ''exp'' are entire functions of genus ''1''.
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| ==Other examples==
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| 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 behaviour 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 theorem of Paley and Wiener, [[Fourier transform]]s of functions with bounded support are entire functions or order ''1'' and finite type.
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| 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]].
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| 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>.
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| 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
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| 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 ''z<sub>n</sub>'' are real, ''p'' ≤ 1, and ''P''(''z'') = ''a'' + ''bz'' + ''cz''<sup>2</sup>, where ''b'' and ''c'' are real, and ''c'' ≤ 0. For example, the sequence of polynomials <math>(1-(z-d)^2/n)^n</math> converges, as ''n'' increases, to exp(−(''z''−''d'')<sup>2</sup>). The polynomials <math>((1+iz/n)^n+(1-iz/n)^n)/2</math> have all real roots, and converge to cos(''z''). Interestingly, the polynomials <math> \prod_{m=1}^n \left(1-\frac{z^2}{((m-\frac{1}{2})\pi)^2}\right)</math> also converge to cos(''z''), showing the buildup of the Hadamard product for cosine.
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| ==See also==
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| * [[Jensen's formula]]
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| * [[Carlson's theorem]]
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| * [[Exponential type]]
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| * [[Paley–Wiener theorem]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{cite book | author = Ralph P. Boas | authorlink = Ralph P. Boas, Jr. | title = Entire Functions | publisher = Academic Press | year = 1954 | id=OCLC [http://worldcat.org/oclc/847696 847696] }}
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| *{{cite book | author = B. Ya. Levin | title = Distribution of zeros of entire functions |
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| publisher = Amer. Math. Soc. |year = 1980}}
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| *{{cite book |author = B. Ya. Levin | title = Lectures on entire functions |
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| publisher = Amer. Math. Soc. |year = 1996}}
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| [[Category:Analytic functions]]
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| [[Category:Special functions]]
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