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)