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

which converges in the whole complex plane, .

If everywhere, then , where is an entire function. If there are finitely many points at which vanishes and these points are (they are called the zeros of the function), then

where is an entire function.

In the general case when has infinitely many zeros there is a product representation (see Weierstrass theorem on infinite products)

(1)

where is an entire function, if , and is the multiplicity of the zero if .

Let

If for large the quantity grows no faster than , then is a polynomial of degree not exceeding . Consequently, if is not a polynomial, then grows faster than any power of . To estimate the growth of in this case one takes as a comparison function the exponential function.

By definition, is an entire function of finite order if there is a finite number such that

The greatest lower bound of the set of numbers satisfying this condition is called the order of the entire function . The order can be computed by the formula

If of order satisfies the condition

(2)

then one says that is a function of order and of finite type. The greatest lower bound of the set of numbers satisfying this condition is called the type of the entire function . It is determined by the formula

Among the entire functions of finite type one distinguishes entire functions of normal type and of minimal type . If the condition (2) does not hold for any , 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

is said to be of exponential type.

The zeros of an entire function of order have the property

Let be the least integer such that . Then the following product representation holds (see Hadamard theorem on entire functions)

(3)

where is a polynomial of degree not exceeding .

To characterize the growth of an entire function of finite order and finite type along rays, one introduces the quantity

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

If

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

A function of several variables is entire if it is analytic for (). 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 the zeros of 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 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 ) 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)