View source for Borel transform

An integral transform of the type

<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/b/b017/b017190/b0171901.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171902.png" /> is an entire function of exponential type. The Borel transform is a special case of the [[Laplace transform|Laplace transform]]. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171903.png" /> is called the Borel transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171904.png" />. If

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then

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the series converges for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171907.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171908.png" /> is the type of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b0171909.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719010.png" /> be the smallest closed convex set containing all the singularities of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719011.png" />; let

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be the supporting function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719013.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719014.png" /> be the growth indicator function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719015.png" />; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719016.png" />. If in a Borel transform the integration takes place over a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719017.png" />, the corresponding integral will converge in the half-plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719018.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719019.png" /> be a closed contour surrounding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719020.png" />; then

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If additional conditions are imposed, other representations may be deduced from this formula. Thus, consider the class of entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719022.png" /> of exponential type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719023.png" /> for which

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This class is identical with the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719025.png" /> that can be represented as

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017190/b01719027.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Borel, "Leçons sur les series divergentes" , Gauthier-Villars (1928) {{MR|}} {{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></table>



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

====References====
<table><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>