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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357201.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \
 +
\sum _ {k = 0 } ^  \infty  a _ {k} z  ^ {k} ,\ \
 +
a _ {k}  =
 +
\frac{f ^ { ( k) } ( 0) }{k! }
 +
,\ \
 +
k \geq  0,
 +
$$
 +
 
 +
which converges in the whole complex plane,  $  \lim\limits _ {k \rightarrow \infty }  | a _ {k} |  ^ {1/k} = 0 $.
  
which converges in the whole complex plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357202.png" />.
+
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
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357203.png" /> everywhere, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357204.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357205.png" /> is an entire function. If there are finitely many points at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357206.png" /> vanishes and these points are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357207.png" /> (they are called the zeros of the function), then
+
$$
 +
f ( z)  = \
 +
( z - z _ {1} ) \dots ( z - z _ {k} ) e ^ {P ( z) } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357208.png" /></td> </tr></table>
+
where  $  P ( z) $
 +
is an entire function.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e0357209.png" /> 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)
  
In the general case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572010.png" /> has infinitely many zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572011.png" /> 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 +
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\frac{z  ^ {k} }{kz _ {k}  ^ {k} }
 +
\right ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572013.png" /> is an entire function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572014.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572016.png" /> is the multiplicity of the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572017.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572018.png" />.
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572019.png" /></td> </tr></table>
+
$$
 +
M ( r)  = \
 +
\max _ {| z | \leq  r } \
 +
| f ( z) | .
 +
$$
 +
 
 +
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.
  
If for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572020.png" /> the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572021.png" /> grows no faster than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572023.png" /> is a polynomial of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572024.png" />. Consequently, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572025.png" /> is not a polynomial, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572026.png" /> grows faster than any power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572027.png" />. To estimate the growth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572028.png" /> 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
  
By definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572029.png" /> is an entire function of finite order if there is a finite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572030.png" /> such that
+
$$
 +
M ( r)  < e ^ {r  ^  \mu  } ,\ \
 +
r > r _ {0} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572031.png" /></td> </tr></table>
+
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
  
The greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572032.png" /> of the set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572033.png" /> satisfying this condition is called the order of the entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572034.png" />. The order can be computed by the formula
+
$$
 +
\rho  = \
 +
\overline{\lim\limits}\; _ {k \rightarrow \infty } \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572035.png" /></td> </tr></table>
+
\frac{k  \mathop{\rm ln}  k }{ \mathop{\rm ln}  | 1 / {a _ {k} } | }
 +
.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572036.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572037.png" /> satisfies the condition
+
If $  f ( z) $
 +
of order $  \rho $
 +
satisfies the condition
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
M ( r) < \
 +
e ^ {\alpha r  ^  \rho  } ,\ \
 +
\alpha  < \infty ,\ \
 +
r > r _ {0} ,
 +
$$
  
then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572039.png" /> is a function of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572040.png" /> and of finite type. The greatest lower bound <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572041.png" /> of the set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572042.png" /> satisfying this condition is called the type of the entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572043.png" />. It is determined by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572044.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {k \rightarrow \infty } \
 +
k ^ { {1 / \rho } }
 +
| a _ {k} |  ^ {1/k}  = \
 +
( \sigma e \rho ) ^ { {1 / \rho } } .
 +
$$
  
Among the entire functions of finite type one distinguishes entire functions of normal type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572045.png" /> and of minimal type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572046.png" />. If the condition (2) does not hold for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572047.png" />, 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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572048.png" /></td> </tr></table>
+
$$
 +
\overline{\lim\limits}\; _ {k \rightarrow \infty }  k
 +
| a _ {k} |  ^ {1/k}  = \
 +
\beta  < \infty ,
 +
$$
  
is said to be of exponential type.
+
is said to be a [[function of exponential type]].
  
The zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572049.png" /> of an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572050.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572051.png" /> have the property
+
The zeros $  z _ {1} , z _ {2} \dots $
 +
of an entire function $  f ( z) $
 +
of order $  \rho $
 +
have the property
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572052.png" /></td> </tr></table>
+
$$
 +
\sum _ {k = 1 } ^  \infty 
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572053.png" /> be the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572054.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572055.png" />. Then the following product representation holds (see [[Hadamard theorem|Hadamard theorem]] on entire functions)
+
\frac{1}{| z _ {k} | ^ {\rho + \epsilon } }
 +
  < \infty ,\ \
 +
\textrm{ for }  \textrm{ all } \
 +
\epsilon > 0.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
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)
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572057.png" /> is a polynomial of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572058.png" />.
+
$$ \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} }
  
To characterize the growth of an entire function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572059.png" /> of finite order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572060.png" /> and finite type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572061.png" /> along rays, one introduces the quantity
+
+ \dots +
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572062.png" /></td> </tr></table>
+
\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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572063.png" /></td> </tr></table>
+
$$
 +
| f ( re ^ {i \phi } ) |  < \
 +
e ^ {( h ( \phi ) + \epsilon ) r  ^  \rho  } ,\ \
 +
r > r _ {0} ( \epsilon ),\ \
 +
\textrm{ for }  \textrm{ all } \
 +
\epsilon > 0.
 +
$$
  
 
If
 
If
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572064.png" /></td> </tr></table>
+
$$
 +
| f ( re ^ {i \phi } ) |  > \
 +
e ^ {( h ( \phi ) - \epsilon ) r  ^  \rho  } ,\ \
 +
r > r _ {0} ( \epsilon ),\ \
 +
z \notin E _ {0} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572065.png" /> is a set which is small in a certain sense (a set of relative measure 0), then the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572066.png" /> are distributed in the plane very regularly in a certain sense, and there is a precise relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572067.png" /> and the characteristic (the density) of the zeros. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572068.png" /> with this property is said to be a function of completely regular growth.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572069.png" /> is entire if it is analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572070.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572071.png" />). 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572072.png" /> the zeros of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572073.png" /> are not isolated.
+
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 &amp; 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 &amp; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572074.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572075.png" />) is also called the Hadamard product representation.
+
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]]].

Latest revision as of 20:13, 10 January 2021


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)
How to Cite This Entry:
Entire function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Entire_function&oldid=14495
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article