View source for Function of exponential type

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An [[entire function]] $f(z)$ satisfying the condition

$$ 
| f ( z) |  <  A e ^ {a | z| } ,\ \ 
| z | < \infty ,\  A , a < \infty .
$$

If  $  f ( z) $
is represented by a series

$$ 
f ( z)  =  \sum _ { k=0 } ^  \infty  \frac{a _ {k} }{k!} z  ^ {k} ,
$$
then
$$ 
\limsup _ {k \rightarrow \infty }  {| a _ {k} | }  ^ {1/k}  <  \infty .
$$

The simplest examples of functions of exponential type are  $  e  ^ {cx} $, 
$  \sin  \alpha z $, 
$  \cos  \beta z $, 
and  $  \sum _ {k=1}  ^ {n} A _ {k} e ^ {a _ {k} z } $.

A function of exponential type has an integral representation

$$ 
f ( z)  =  
\frac{1}{2 \pi i }
 \int\limits _ { C }
\gamma ( t) e  ^ {zt}  d t ,
$$

where  $  \gamma ( t) $
is the function associated with  $  f ( z) $
in the sense of Borel (see [[Borel transform]]) and  $  C $
is a closed contour enclosing all the singularities of  $  \gamma ( t) $.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) {{MR|0156975}} {{ZBL|0152.06703}} </TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R.P. Boas, "Entire functions" , Acad. Press (1954) {{MR|0068627}} {{ZBL|0058.30201}} </TD></TR>
</table>