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An integral transform of the type

$$ 
\gamma (t)  =  \int\limits _ { 0 } ^  \infty  
f(z)e  ^ {-zt}  dz,
$$

where  $  f(z) $
is an entire [[function of exponential type]]. The Borel transform is a special case of the [[Laplace transform|Laplace transform]]. The function  $  \gamma (t) $
is called the Borel transform of  $  f(z) $. 
If

$$ 
f(z)  =  \sum _ { n=0 } ^  \infty  

\frac{a _ {n} }{n!}
 z  ^ {n} ,
$$

then

$$ 
\gamma (t)  =  \sum _ { v=0 } ^  \infty  
a _ {v} t ^ {-(v+1) } ;
$$

the series converges for  $  | t | > \sigma $, 
where  $  \sigma $
is the type of  $  f(z) $. 
Let  $  \overline{D}\; $
be the smallest closed convex set containing all the singularities of the function  $  \gamma (t) $; 
let

$$ 
K( \phi )  =  \max _ {z \in \overline{D}\; } \ 
 \mathop{\rm Re} (ze ^ {-i \phi } )
$$

be the supporting function of  $  \overline{D}\; $; 
and let  $  h ( \phi ) $
be the growth indicator function of  $  f(z) $; 
then  $  K( \phi ) = h( - \phi ) $. 
If in a Borel transform the integration takes place over a ray  $   \mathop{\rm arg}  z = \phi $, 
the corresponding integral will converge in the half-plane  $  x  \cos  \phi + y  \sin  \phi > K ( - \phi ) $. 
Let  $  C $
be a closed contour surrounding  $  \overline{D}\; $; 
then

$$ 
f(z)  =  
\frac{1}{2 \pi i }

\int\limits _ { C } \gamma (t) e  ^ {zt}  dt.
$$

If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions  $  f(z) $
of exponential type  $  \leq  \sigma $
for which

$$ 
\int\limits _ {- \infty } ^  \infty  
| f(x) |  ^ {2}  dx  <  \infty .
$$

This class is identical with the class of functions  $  f(z) $
that can be represented as

$$ 
f(z)  = \ 

\frac{1}{\sqrt {2 \pi } }

\int\limits _ {- \sigma } ^  \sigma  
e  ^ {izt} \phi (t)  dt,
$$

where  $  \phi (t) \in {L _ {2} } ( - \sigma , \sigma ) $.

====Comments====
The statement at the end of the article above is called the [[Paley–Wiener theorem]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> E. Borel, "Leçons sur les séries divergentes" , Gauthier-Villars (1928) {{ZBL|54.0223.01}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> M.M. Dzhrbashyan, "Integral transforms and representation of functions in the complex domain" , Moscow (1966) (In Russian)</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>