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''of a complex variable''
 
''of a complex variable''
  
A function which is obtained in some manner from a given function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a0134901.png" /> with the aid of some fixed function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a0134902.png" />. For example, if
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A function which is obtained in some manner from a given function $  f(z) $
 +
with the aid of some fixed function $  F(z) $.  
 +
For example, if
  
<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/a/a013/a013490/a0134903.png" /></td> </tr></table>
+
$$
 +
f (z)  = \sum _ {k=0 } ^  \infty  a _ {k} z  ^ {k}
 +
$$
  
 
is an entire function and if
 
is an entire function and if
  
<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/a/a013/a013490/a0134904.png" /></td> </tr></table>
+
$$
 +
F (z)  = \sum _ {k=0 } ^  \infty  b _ {k} z  ^ {k}
 +
$$
  
is a fixed entire function with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a0134905.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a0134906.png" />, then
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is a fixed entire function with $  b _ {k} \neq 0 $,  
 +
$  k \geq  0 $,  
 +
then
  
<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/a/a013/a013490/a0134907.png" /></td> </tr></table>
+
$$
 +
\gamma (z)  = \sum _ {k=0 } ^  \infty 
 +
a _
 +
\frac{k}{b} _ {k} z  ^ {-(k+1)}
 +
$$
  
is a function which is associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a0134908.png" /> by means of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a0134909.png" />; it is assumed that the series converges in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a01349010.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a01349011.png" /> is then represented in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a01349012.png" /> by the formula
+
is a function which is associated to $  f(z) $
 +
by means of the function $  F(z) $;  
 +
it is assumed that the series converges in some neighbourhood $  | z | > R $.  
 +
The function $  f(z) $
 +
is then represented in terms of $  \gamma (z) $
 +
by the formula
  
<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/a/a013/a013490/a01349013.png" /></td> </tr></table>
+
$$
 +
f (z)  =
 +
\frac{1}{2 \pi i }
 +
\int\limits _ {| t | = R _ {1} > R }
 +
\gamma (t) F (zt)  dt .
 +
$$
  
 
In particular, if
 
In particular, if
  
<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/a/a013/a013490/a01349014.png" /></td> </tr></table>
+
$$
 +
f (z)  = \sum _ {k=0 } ^  \infty 
 +
 
 +
\frac{a ^ {k} }{k!}
 +
z  ^ {k}
 +
$$
  
is an entire function of exponential type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a01349015.png" />, then
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is an entire function of exponential type and $  F(z) = e  ^ {z} $,  
 +
then
  
<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/a/a013/a013490/a01349016.png" /></td> </tr></table>
+
$$
 +
\gamma (z)  = \sum _ {k=0 } ^  \infty 
 +
a _ {k} z  ^ {-(k+1)}
 +
$$
  
is the Borel-associated function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013490/a01349017.png" /> (cf. [[Borel transform|Borel transform]]).
+
is the Borel-associated function of $  f(z) $(
 +
cf. [[Borel transform|Borel transform]]).

Revision as of 18:48, 5 April 2020


of a complex variable

A function which is obtained in some manner from a given function $ f(z) $ with the aid of some fixed function $ F(z) $. For example, if

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

is an entire function and if

$$ F (z) = \sum _ {k=0 } ^ \infty b _ {k} z ^ {k} $$

is a fixed entire function with $ b _ {k} \neq 0 $, $ k \geq 0 $, then

$$ \gamma (z) = \sum _ {k=0 } ^ \infty a _ \frac{k}{b} _ {k} z ^ {-(k+1)} $$

is a function which is associated to $ f(z) $ by means of the function $ F(z) $; it is assumed that the series converges in some neighbourhood $ | z | > R $. The function $ f(z) $ is then represented in terms of $ \gamma (z) $ by the formula

$$ f (z) = \frac{1}{2 \pi i } \int\limits _ {| t | = R _ {1} > R } \gamma (t) F (zt) dt . $$

In particular, if

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

is an entire function of exponential type and $ F(z) = e ^ {z} $, then

$$ \gamma (z) = \sum _ {k=0 } ^ \infty a _ {k} z ^ {-(k+1)} $$

is the Borel-associated function of $ f(z) $( cf. Borel transform).

How to Cite This Entry:
Associated function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Associated_function&oldid=45229
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