Entire function

A function that is analytic in the whole complex plane (except, possibly, at the point at infinity). It can be expanded in a power series

$$f ( z) = \ \sum _ {k = 0 } ^ \infty a _ {k} z ^ {k} ,\ \ a _ {k} = \frac{f ^ { ( k) } ( 0) }{k! } ,\ \ k \geq 0,$$

which converges in the whole complex plane, $\lim\limits _ {k \rightarrow \infty } | a _ {k} | ^ {1/k} = 0$.

If $f ( z) \neq 0$ everywhere, then $f ( z) = e ^ {P ( z) }$, where $P ( z)$ is an entire function. If there are finitely many points at which $f ( z)$ vanishes and these points are $z _ {1} \dots z _ {k}$( they are called the zeros of the function), then

$$f ( z) = \ ( z - z _ {1} ) \dots ( z - z _ {k} ) e ^ {P ( z) } ,$$

where $P ( z)$ is an entire function.

In the general case when $f ( z)$ has infinitely many zeros $z _ {1} , z _ {2} \dots$ there is a product representation (see Weierstrass theorem on infinite products)

$$\tag{1 } f ( z) = \ z ^ \lambda e ^ {P ( z) } \prod _ {k = 1 } ^ \infty \left ( 1 - \frac{z}{z _ {k} } \right ) \mathop{\rm exp} \left ( { \frac{z}{z _ {k} } } + \dots + \frac{z ^ {k} }{kz _ {k} ^ {k} } \right ) ,$$

where $P ( z)$ is an entire function, $\lambda = 0$ if $f ( 0) \neq 0$, and $\lambda$ is the multiplicity of the zero $z = 0$ if $f ( 0) = 0$.

Let

$$M ( r) = \ \max _ {| z | \leq r } \ | f ( z) | .$$

If for large $r$ the quantity $M ( r)$ grows no faster than $r ^ \mu$, then $f ( z)$ is a polynomial of degree not exceeding $\mu$. Consequently, if $f ( z)$ is not a polynomial, then $M ( r)$ grows faster than any power of $r$. To estimate the growth of $M ( r)$ in this case one takes as a comparison function the exponential function.

By definition, $f ( z)$ is an entire function of finite order if there is a finite number $\mu$ such that

$$M ( r) < e ^ {r ^ \mu } ,\ \ r > r _ {0} .$$

The greatest lower bound $\rho$ of the set of numbers $\mu$ satisfying this condition is called the order of the entire function $f ( z)$. The order can be computed by the formula

$$\rho = \ \overline{\lim\limits}\; _ {k \rightarrow \infty } \ \frac{k \mathop{\rm ln} k }{ \mathop{\rm ln} | 1 / {a _ {k} } | } .$$

If $f ( z)$ of order $\rho$ satisfies the condition

$$\tag{2 } M ( r) < \ e ^ {\alpha r ^ \rho } ,\ \ \alpha < \infty ,\ \ r > r _ {0} ,$$

then one says that $f ( z)$ is a function of order $\rho$ and of finite type. The greatest lower bound $\sigma$ of the set of numbers $\alpha$ satisfying this condition is called the type of the entire function $f ( z)$. It is determined by the formula

$$\overline{\lim\limits}\; _ {k \rightarrow \infty } \ k ^ { {1 / \rho } } | a _ {k} | ^ {1/k} = \ ( \sigma e \rho ) ^ { {1 / \rho } } .$$

Among the entire functions of finite type one distinguishes entire functions of normal type $( \sigma > 0)$ and of minimal type $( \sigma = 0)$. If the condition (2) does not hold for any $\alpha < \infty$, then the function is said to be an entire function of maximal type or of infinite type. An entire function of order 1 and of finite type, and also an entire function of order less than 1, characterized by the condition

$$\overline{\lim\limits}\; _ {k \rightarrow \infty } k | a _ {k} | ^ {1/k} = \ \beta < \infty ,$$

is said to be a function of exponential type.

The zeros $z _ {1} , z _ {2} \dots$ of an entire function $f ( z)$ of order $\rho$ have the property

$$\sum _ {k = 1 } ^ \infty \frac{1}{| z _ {k} | ^ {\rho + \epsilon } } < \infty ,\ \ \textrm{ for } \textrm{ all } \ \epsilon > 0.$$

Let $p$ be the least integer $( p \leq \rho )$ such that $\sum _ {k = 1 } ^ \infty | z _ {k} | ^ {- p - 1 } < \infty$. Then the following product representation holds (see Hadamard theorem on entire functions)

$$\tag{3 } f ( z) = \ z ^ \lambda e ^ {P ( z) } \prod _ {k = 1 } ^ \infty \left ( 1 - \frac{z}{z _ {k} } \right ) \mathop{\rm exp} \left ( \frac{z}{z _ {k} } + \dots + \frac{z ^ {p} }{pz _ {k} ^ {p} } \right ) ,$$

where $P ( z)$ is a polynomial of degree not exceeding $\rho$.

To characterize the growth of an entire function $f ( z)$ of finite order $\rho$ and finite type $\sigma$ along rays, one introduces the quantity

$$h ( \phi ) = \ \overline{\lim\limits}\; _ {r \rightarrow \infty } \ \frac{ \mathop{\rm ln} | f ( re ^ {i \phi } ) | }{r ^ \rho }$$

— the growth indicator (cf. Growth indicatrix). Here, one always has

$$| f ( re ^ {i \phi } ) | < \ e ^ {( h ( \phi ) + \epsilon ) r ^ \rho } ,\ \ r > r _ {0} ( \epsilon ),\ \ \textrm{ for } \textrm{ all } \ \epsilon > 0.$$

If

$$| f ( re ^ {i \phi } ) | > \ e ^ {( h ( \phi ) - \epsilon ) r ^ \rho } ,\ \ r > r _ {0} ( \epsilon ),\ \ z \notin E _ {0} ,$$

where $E _ {0}$ is a set which is small in a certain sense (a set of relative measure 0), then the zeros of $f ( z)$ are distributed in the plane very regularly in a certain sense, and there is a precise relation between $h ( \phi )$ and the characteristic (the density) of the zeros. A function $f ( z)$ with this property is said to be a function of completely regular growth.

A function of several variables $f ( z _ {1} \dots z _ {n} )$ is entire if it is analytic for $| z _ {k} | < \infty$( $k = 1 \dots n$). Again one may introduce the concepts of order and type (conjugate orders and types). A simple representation in the form of an infinite product is not available here, because in contrast to the case $n = 1$ the zeros of $f ( z)$ are not isolated.

References

[1]	M.A. Evgrafov, "Asymptotic estimates and entire functions" , Gordon & Breach , Moscow (1979) (In Russian)
[2]	B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian)
[3]	L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , Transl. Math. Monogr. , 44 , Amer. Math. Soc. (1974) (Translated from Russian)

Comments

The "product representation" (1) mentioned above (when $f ( z)$ has infinitely many zeros) is also called the Weierstrass product representation. The representation (3) (in which the polynomials occurring in the exponent are of fixed degree $p$) is also called the Hadamard product representation.

Entire functions are sometimes, especially in older literature, called integral functions, cf. [a2], [a3]. An elementary account is [a4]. For (analogues of Hadamard's theorem for) entire functions of several complex variables see [3], [a5]. For distribution of zeros and related matters in one variable cf. [2], [a7].

References

[a1]	R.P. Boas, "Entire functions" , Acad. Press (1954)
[a2]	M.L. Cartwright, "Integral functions" , Cambridge Univ. Press (1962)
[a3]	G. Valiron, "Lectures on the general theory of integral functions" , Chelsea (1949) (Translated from French)
[a4]	A.S.B. Holland, "Introduction to the theory of entire functions" , Acad. Press (1973)
[a5]	P. Lelong, L. Gruman, "Entire functions of several complex variables" , Springer (1986)
[a6]	P. Lelong, "Fonctionelles analytiques et fonctions entières ( variables)" , Univ. Montréal (1968)
[a7]	N. Levinson, "Gap and density theorems" , Amer. Math. Soc. (1968)