Entire function

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 of an entire function ${\displaystyle f(z)}$ is defined using the limit superior as$${\displaystyle \rho =\underset{r\to \mathrm{\infty }}{lim sup}\frac{{\log }^{+}{\log }^{+}M(r,f)}{\log r},}$$where ${\displaystyle {\log }^{+}x=max(0,\log x)}$ and$${\displaystyle M(r,f)=\underset{|z|=r}{sup}|f(z)|}$$is the maximum modulus function. The order is a non-negative real number or infinity. In other words, the order of ${\displaystyle f(z)}$ is the infimum of all ${\displaystyle m}$ such that:$${\displaystyle f(z)=O(e^{|z{|}^m}),\text{as }z\to \mathrm{\infty }.}$$The example of ${\displaystyle f(z)=z}$ shows that the infimum is generally not a minimum, that is ${\displaystyle f(z)\ne O(e^{|z{|}^{\rho }})}$.

If ⁠${\displaystyle 0<\rho <\mathrm{\infty }}$⁠, one can also define the type:$${\displaystyle \sigma =\underset{r\to \mathrm{\infty }}{lim sup}\frac{\log M(r,f)}{r^{\rho }}.}$$If the order is ${\displaystyle 1}$ and the type is ⁠${\displaystyle \sigma }$⁠, the function is said to be "of exponential type ⁠${\displaystyle \sigma }$⁠". If it is of order less than ${\displaystyle 1}$ it is said to be of exponential type ${\displaystyle 0}$.

If $${\displaystyle f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n,}$$then the order and type can be found by the formulas $${\displaystyle \begin{array}{cc}\rho & =\underset{n\to \mathrm{\infty }}{lim sup}\frac{n\ln n}{-\ln |a_n|}\\ (e\rho \sigma {)}^{\frac{1}{\rho }}& =\underset{n\to \mathrm{\infty }}{lim sup}{n}^{\frac{1}{\rho }}|a_n{|}^{\frac{1}{n}}\end{array}}$$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 functions ${\displaystyle J_n(z)}$ and spherical Bessel functions ${\displaystyle j_n(z)}$ for integer values of ${\displaystyle n}$
• 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 ${\displaystyle g=\lfloor \rho \rfloor }$ 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.