Difference between revisions of "Entire function"
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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 | 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|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 | + | 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 | 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 | + | 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 | + | 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 | + | 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 of exponential type. | is said to be of exponential type. | ||
− | The zeros | + | 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|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|Growth indicatrix]]). Here, one always has | — the growth indicator (cf. [[Growth indicatrix|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 | If | ||
− | + | $$ | |
+ | | f ( re ^ {i \phi } ) | > \ | ||
+ | e ^ {( h ( \phi ) - \epsilon ) r ^ \rho } ,\ \ | ||
+ | r > r _ {0} ( \epsilon ),\ \ | ||
+ | z \notin E _ {0} , | ||
+ | $$ | ||
− | where | + | 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Evgrafov, "Asymptotic estimates and entire functions" , Gordon & Breach , Moscow (1979) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc. (1974) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Evgrafov, "Asymptotic estimates and entire functions" , Gordon & Breach , Moscow (1979) (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1980) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.I. Ronkin, "Inroduction to the theory of entire functions of several variables" , ''Transl. Math. Monogr.'' , '''44''' , Amer. Math. Soc. (1974) (Translated from Russian)</TD></TR></table> | ||
− | |||
− | |||
====Comments==== | ====Comments==== | ||
− | The "product representation" (1) mentioned above (when | + | 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. [[#References|[a2]]], [[#References|[a3]]]. An elementary account is [[#References|[a4]]]. For (analogues of Hadamard's theorem for) entire functions of several complex variables see [[#References|[3]]], [[#References|[a5]]]. For distribution of zeros and related matters in one variable cf. [[#References|[2]]], [[#References|[a7]]]. | Entire functions are sometimes, especially in older literature, called integral functions, cf. [[#References|[a2]]], [[#References|[a3]]]. An elementary account is [[#References|[a4]]]. For (analogues of Hadamard's theorem for) entire functions of several complex variables see [[#References|[3]]], [[#References|[a5]]]. For distribution of zeros and related matters in one variable cf. [[#References|[2]]], [[#References|[a7]]]. |
Revision as of 19:37, 5 June 2020
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 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 (![]() |
[a7] | N. Levinson, "Gap and density theorems" , Amer. Math. Soc. (1968) |
Entire function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Entire_function&oldid=46825