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An [[Entire function|entire function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f0419901.png" /> satisfying the condition
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$#C+1 = 14 : ~/encyclopedia/old_files/data/F041/F.0401990 Function of exponential type
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<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/f/f041/f041990/f0419902.png" /></td> </tr></table>
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{{TEX|auto}}
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{{TEX|done}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f0419903.png" /> is represented by a series
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An [[Entire function|entire function]]  $  f ( z) $
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satisfying 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/f/f041/f041990/f0419904.png" /></td> </tr></table>
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$$
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| f ( z) |  < A e ^ {a | z| } ,\ \
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| z | < \infty ,\  A , a < \infty .
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$$
  
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If  $  f ( z) $
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is represented by a series
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$$
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f ( z)  =  \sum _ { k=0 } ^  \infty  \frac{a _ {k} }{k!} z  ^ {k} ,
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$$
 
then
 
then
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$$
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\limsup _ {k \rightarrow \infty }  {| a _ {k} | }  ^ {1/k}  <  \infty .
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$$
  
<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/f/f041/f041990/f0419905.png" /></td> </tr></table>
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The simplest examples of functions of exponential type are $  e  ^ {cx} $,  
 
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$  \sin  \alpha z $,  
The simplest examples of functions of exponential type are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f0419906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f0419907.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f0419908.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f0419909.png" />.
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$  \cos  \beta z $,  
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and $  \sum _ {k=1}  ^ {n} A _ {k} e ^ {a _ {k} z } $.
  
 
A function of exponential type has an integral representation
 
A function of exponential type has an integral representation
  
<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/f/f041/f041990/f04199010.png" /></td> </tr></table>
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$$
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f ( z)  =
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\frac{1}{2 \pi i }
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\int\limits _ { C }
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\gamma ( t) e  ^ {zt}  d t ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f04199011.png" /> is the function associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f04199012.png" /> in the sense of Borel (see [[Borel transform|Borel transform]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f04199013.png" /> is a closed contour enclosing all the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041990/f04199014.png" />.
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where $  \gamma ( t) $
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is the function associated with f ( z) $
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in the sense of Borel (see [[Borel transform|Borel transform]]) and $  C $
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is a closed contour enclosing all the singularities of $  \gamma ( t) $.
  
 
====References====
 
====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)</TD></TR></table>
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<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></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.P. Boas,   "Entire functions" , Acad. Press (1954)</TD></TR></table>
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<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>

Revision as of 20:07, 10 January 2021


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

[1] B.Ya. Levin, "Distribution of zeros of entire functions" , Amer. Math. Soc. (1964) (Translated from Russian) MR0156975 Zbl 0152.06703

Comments

References

[a1] R.P. Boas, "Entire functions" , Acad. Press (1954) MR0068627 Zbl 0058.30201
How to Cite This Entry:
Function of exponential type. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Function_of_exponential_type&oldid=12134
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