Movatterモバイル変換

Jump to content

Entire function

From Wikipedia, the free encyclopedia

Function that is holomorphic on the whole complex plane

Incomplex analysis, anentire function, also called anintegral function, is a complex-valuedfunction that isholomorphic on the wholecomplex plane. Typical examples of entire functions arepolynomials and theexponential function, and any finite sums, products and compositions of these, such as thetrigonometric functions sine andcosine and theirhyperbolic counterparts sinh andcosh, as well asderivatives andintegrals of entire functions such as theerror function. If an entire function $f(z)$ has aroot at⁠ $w {\displaystyle w}$ ⁠, then⁠ $f(z)/(z-w)$ ⁠, taking the limit value at⁠ $w {\displaystyle w}$ ⁠, is an entire function. On the other hand, thenatural logarithm, thereciprocal function, and thesquare root are all not entire functions, nor can they becontinued analytically to an entire function.

Atranscendental entire function is an entire function that is not a polynomial.

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

Properties

Every entire function $f(z)$ can be represented as a singlepower series: $\ f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\$ thatconverges everywhere in the complex plane, henceuniformly on compact sets. Theradius of convergence is infinite, which implies that $\ \lim _{n\to \infty }|a_{n}|^{\frac {1}{n}}=0\$ or, equivalently,^[a] $\ \lim _{n\to \infty }{\frac {\ln |a_{n}|}{n}}=-\infty ~.$ 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 thecomplex conjugate of $z {\displaystyle z}$ will be the complex conjugate of the value at⁠ $z {\displaystyle z}$ ⁠. Such functions are sometimes called self-conjugate (the conjugate function,⁠ $F^{*}(z)$ ⁠, being given by⁠ ${\bar {F}}({\bar {z}})$ ⁠).^[1]

If the real part of an entire function is known in a (complex) 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 aneighborhood of zero, then we can find the coefficients for $n>0$ from the following derivatives with respect to a real variable⁠ $r {\displaystyle r}$ ⁠: ${\begin{aligned}\operatorname {\mathcal {Re}} \left\{\ a_{n}\ \right\}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\ \operatorname {\mathcal {Re}} \left\{\ f(r)\ \right\}&&\quad \mathrm {at} \quad r=0\\\operatorname {\mathcal {Im}} \left\{\ a_{n}\ \right\}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\ \operatorname {\mathcal {Re}} \left\{\ f\left(r\ e^{-{\frac {i\pi }{2n}}}\right)\ \right\}&&\quad \mathrm {at} \quad r=0\end{aligned}}$

(Likewise, if the imaginary part is known in such 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.^[b]Note however that an entire function isnot necessarily determined by its real part on some other 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 $i {\displaystyle i}$ 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 someimaginary number.

TheWeierstrass factorization theorem asserts that any entire function can be represented by a product involving itszeroes (or "roots").

The entire functions on the complex plane form anintegral domain (in fact aPrüfer domain). They also form acommutative unital associative algebra over thecomplex numbers.

Liouville's theorem states that anybounded entire function must be constant.^[c]

As a consequence of Liouville's theorem, any function that is entire on the wholeRiemann sphere^[d]is constant. Thus any non-constant entire function must have asingularity at the complexpoint at infinity, either apole for a polynomial or anessential singularity for atranscendental entire function. Specifically, by theCasorati–Weierstrass theorem, for any transcendental entire function $f {\displaystyle f}$ and any complex $w {\displaystyle w}$ there is asequence $(z_{m})_{m\in \mathbb {N} }$ such that

\ \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 alacunary value of the function. The possibility of a lacunary value is illustrated by theexponential function, which never takes on the value⁠ $0 {\displaystyle 0}$ ⁠. One can take a suitable branch of the logarithm of an entire function that never hits⁠ $0 {\displaystyle 0}$ ⁠, so that this will also be an entire function (according to theWeierstrass factorization theorem). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than $0 {\displaystyle 0}$ 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 $M {\displaystyle M}$ ⁠, $R {\displaystyle R}$ are positive constants and $n {\displaystyle n}$ is a non-negative integer. An entire function $f {\displaystyle f}$ satisfying the inequality $|f(z)|\leq M|z|^{n}$ for all $z {\displaystyle z}$ with⁠ $\vert z$ ⁠, is necessarily a polynomial, ofdegree at most⁠ $n {\displaystyle n}$ ⁠.^[e]Similarly, an entire function $f {\displaystyle f}$ satisfying the inequality $M|z|^{n}\leq |f(z)|$ for all $z {\displaystyle z}$ with⁠ $\vert z\vert \geq R$ ⁠, is necessarily a polynomial, of degree at least⁠ $n {\displaystyle n}$ ⁠.

Growth

Entire functions may grow as fast as any increasing function: for any increasing function $g:[0,\infty )\to [0,\infty )$ there exists an entire function $f {\displaystyle f}$ such that $f(x)>g(|x|)$ for all real⁠ $x {\displaystyle x}$ ⁠. Such a function $f {\displaystyle f}$ may be easily found of the form: $f(z)=c+\sum _{k=1}^{\infty }\left({\frac {z}{k}}\right)^{n_{k}}$ for a constant $c {\displaystyle c}$ and a strictly increasing sequence of positive integers⁠ $n_{k}$ ⁠. 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 $x {\displaystyle x}$ ⁠. (For instance, it certainly holds if one chooses $c:=g(2)$ and, for any integer $k\geq 1$ one chooses an even exponent $n_{k}$ such that⁠ $\left({\frac {k+1}{k}}\right)^{n_{k}}\geq g(k+2)$ ⁠).

Order and type

Theorder (at infinity) of an entire function $f(z)$ is defined using thelimit superior as: $\rho =\limsup _{r\to \infty }{\frac {\ln \left(\ln \|f\|_{\infty ,B_{r}}\right)}{\ln r}},$ where $B_{r}$ is the disk of radius $r {\displaystyle r}$ and $\|f\|_{\infty ,B_{r}}$ denotes thesupremum norm of $f(z)$ on⁠ $B_{r}$ ⁠. The order is a non-negativereal number or infinity (except when $f(z)=0$ for all⁠ $z {\displaystyle z}$ ⁠). In other words, the order of $f(z)$ is theinfimum of all $m {\displaystyle m}$ such that: $f(z)=O\left(\exp \left(|z|^{m}\right)\right),\quad {\text{as }}z\to \infty .$

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

If⁠ $0<\rho <\infty$ ⁠, one can also define thetype: $\sigma =\limsup _{r\to \infty }{\frac {\ln \|f\|_{\infty ,B_{r}}}{r^{\rho }}}.$

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

If $f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},$ then the order and type can be found by the formulas ${\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{-\ln |a_{n}|}}\\[6pt](e\rho \sigma )^{\frac {1}{\rho }}&=\limsup _{n\to \infty }n^{\frac {1}{\rho }}|a_{n}|^{\frac {1}{n}}\end{aligned}}$

Let $f^{(n)}$ denote the⁠ $n {\displaystyle n}$ ⁠th derivative of⁠ $f {\displaystyle f}$ ⁠. Then we may restate these formulas in terms of the derivatives at any arbitrary point⁠ $z_{0}$ ⁠: ${\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{n\ln n-\ln |f^{(n)}(z_{0})|}}=\left(1-\limsup _{n\to \infty }{\frac {\ln |f^{(n)}(z_{0})|}{n\ln n}}\right)^{-1}\\[6pt](\rho \sigma )^{\frac {1}{\rho }}&=e^{1-{\frac {1}{\rho }}}\limsup _{n\to \infty }{\frac {|f^{(n)}(z_{0})|^{\frac {1}{n}}}{n^{1-{\frac {1}{\rho }}}}}\end{aligned}}$

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

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

Examples

Here are some examples of functions of various orders:

Orderρ

For arbitrary positive numbers $\rho$ and $\sigma$ one can construct an example of an entire function of order $\rho$ and type $\sigma$ using: $f(z)=\sum _{n=1}^{\infty }\left({\frac {e\rho \sigma }{n}}\right)^{\frac {n}{\rho }}z^{n}$

Order 0

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

Order 1/4

$f({\sqrt[{4}]{z}}),$ where $f(u)=\cos(u)+\cosh(u)$

Order 1/3

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

Order 1/2

$\cos \left(a{\sqrt {z}}\right)$ with $a\neq 0$ (for which the type is given by⁠ $\sigma =\vert a\vert$ ⁠)

Order 1

$\exp(az)$ with $a\neq 0$ (⁠ $\sigma =\vert a\vert$ ⁠)
$\sin(z)$
$\cosh(z)$
theBessel functions $J_{n}(z)$ and spherical Bessel functions $j_{n}(z)$ for integer values of $n {\displaystyle n}$ ^[2]
thereciprocal gamma function $1/\Gamma (z)$ (⁠ $\sigma$ ⁠ is infinite)
$\sum _{n=2}^{\infty }{\frac {z^{n}}{(n\ln n)^{n}}}\quad (\sigma =0)$

Order 3/2

Airy function $Ai(z)$

Order 2

$\exp(az^{2})$ with $a\neq 0$ (⁠ $\sigma =\vert a\vert$ ⁠)
TheBarnes G-function (⁠ $\sigma$ ⁠ is infinite)

Order infinity

$\exp(\exp(z))$

Genus

Entire functions of finite order haveHadamard's canonical representation (Hadamard factorization theorem): $f(z)=z^{m}e^{P(z)}\prod _{n=1}^{\infty }\left(1-{\frac {z}{z_{n}}}\right)\exp \left({\frac {z}{z_{n}}}+\cdots +{\frac {1}{p}}\left({\frac {z}{z_{n}}}\right)^{p}\right),$ where $z_{k}$ are thoseroots of $f {\displaystyle f}$ that are not zero (⁠ $z_{k}\neq 0$ ⁠), $m {\displaystyle m}$ is the order of the zero of $f {\displaystyle f}$ at $z=0$ (the case $m=0$ being taken to mean⁠ $f(0)\neq 0$ ⁠), $P {\displaystyle P}$ a polynomial (whose degree we shall call $q {\displaystyle q}$ ), and $p {\displaystyle p}$ is the smallest non-negative integer such that the series $\sum _{n=1}^{\infty }{\frac {1}{|z_{n}|^{p+1}}}$ converges. The non-negative integer $g=\max\{p,q\}$ is called the genus of the entire function⁠ $f {\displaystyle f}$ ⁠.

If the order $\rho$ is not an integer, then $g=\lfloor \rho \rfloor$ is the integer part of⁠ $\rho$ ⁠. If the order is a positive integer, then there are two possibilities: $g=\rho -1$ or⁠ $g=\rho$ ⁠.

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

Other examples

According toJ. E. Littlewood, theWeierstrass 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 theFresnel integrals, theJacobi theta function, and thereciprocal Gamma function. The exponential function and the error function are special cases of theMittag-Leffler function. According to the fundamentaltheorem of Paley and Wiener,Fourier transforms of functions (or distributions) with bounded support are entire functions of order $1 {\displaystyle 1}$ 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 andParabolic cylinder functions arise in this way. The class of entire functions is closed with respect to compositions. This makes it possible to studydynamics of entire functions.

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

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 theLaguerre–Pólya class, which can also be characterized in terms of the Hadamard product, namely, $f {\displaystyle f}$ belongs to this classif and only if in the Hadamard representation all $z_{n}$ are real,⁠ $\rho \leq 1$ ⁠, and⁠ $P(z)=a+bz+cz^{2}$ ⁠, where $b {\displaystyle b}$ and $c {\displaystyle c}$ are real, and⁠ $c\leq 0$ ⁠. For example, the sequence of polynomials $\left(1-{\frac {(z-d)^{2}}{n}}\right)^{n}$ converges, as $n {\displaystyle n}$ increases, to⁠ $\exp(-(z-d)^{2})$ ⁠. The polynomials ${\frac {1}{2}}\left(\left(1+{\frac {iz}{n}}\right)^{n}+\left(1-{\frac {iz}{n}}\right)^{n}\right)$ have all real roots, and converge to⁠ $\cos(z)$ ⁠. The polynomials $\prod _{m=1}^{n}\left(1-{\frac {z^{2}}{\left(\left(m-{\frac {1}{2}}\right)\pi \right)^{2}}}\right)$ also converge to⁠ $\cos(z)$ ⁠, showing the buildup of the Hadamard product for cosine.

See also

Notes

^If necessary, the logarithm of zero is taken to be equal to minus infinity.
^For instance, if the real part is known on part of the unit circle, then it is known on the whole unit circle byanalytic extension, and then the coefficients of the infinite series are determined from the coefficients of theFourier series for the real part on the unit circle.
^Liouville's theorem may be used to elegantly prove thefundamental theorem of algebra.
^TheRiemann sphere is the whole complex plane augmented with a single point at infinity.
^The converse is also true as for any polynomial ${\textstyle p(z)=\sum _{k=0}^{n}a_{k}z^{k}}$ of degree $n {\displaystyle n}$ the inequality ${\textstyle |p(z)|\leq \left(\ \sum _{k=0}^{n}|a_{k}|\ \right)|z|^{n}}$ holds for any $|z|\geq 1~.$

References

^Boas 1954, p. 1.
^See asymptotic expansion in Abramowitz and Stegun,p. 377, 9.7.1.

Sources

Boas, Ralph P. (1954).Entire Functions.Academic Press.ISBN 9780080873138.OCLC 847696.{{cite book}}:ISBN / Date incompatibility (help)
Levin, B. Ya. (1980) [1964].Distribution of Zeros of Entire Functions.American Mathematical Society.ISBN 978-0-8218-4505-9.
Levin, B. Ya. (1996).Lectures on Entire Functions.American Mathematical Society.ISBN 978-0-8218-0897-9.

Retrieved from "https://en.wikipedia.org/w/index.php?title=Entire_function&oldid=1325015479"

Hidden categories:

[8]ページ先頭

©2009-2026 Movatter.jp