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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

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

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  $  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|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

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

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====
<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====
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]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Boas,   "Entire functions" , Acad. Press  (1954)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.L. Cartwright,   "Integral functions" , Cambridge Univ. Press  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Valiron,   "Lectures on the general theory of integral functions" , Chelsea  (1949)  (Translated from French)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.S.B. Holland,   "Introduction to the theory of entire functions" , Acad. Press  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Lelong,   L. Gruman,   "Entire functions of several complex variables" , Springer  (1986)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Lelong,   "Fonctionelles analytiques et fonctions entières (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035720/e03572076.png" /> variables)" , Univ. Montréal  (1968)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  N. Levinson,   "Gap and density theorems" , Amer. Math. Soc.  (1968)</TD></TR></table>