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The Euler transformation of series. Given a series
 
The Euler transformation of series. Given a series
  
<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/e/e036/e036620/e0366201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\sum _ { n= } 0 ^  \infty  (- 1) ^ {n} a _ {n} ,
 +
$$
  
 
the series
 
the series
  
<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/e/e036/e036620/e0366202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\sum _ { n= } 0 ^  \infty 
 +
\frac{\Delta  ^ {n} a _ {0} }{2 ^ {n+} 1 }
 +
 
 +
$$
  
 
is said to be obtained from (1) by means of the Euler transformation. Here
 
is said to be obtained from (1) by means of the Euler transformation. Here
  
<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/e/e036/e036620/e0366203.png" /></td> </tr></table>
+
$$
 +
\Delta  ^ {n} a _ {0}  = \sum _ { k= } 0 ^ { n }
 +
(- 1)  ^ {k} \left ( \begin{array}{c}
 +
n \\
 +
k
 +
\end{array}
 +
\right ) a _ {k} .
 +
$$
  
 
If the series (1) converges, then so does (2), and to the same sum as (1). If the series (2) converges (in this case (1) may diverge), then the series (1) is called Euler summable.
 
If the series (1) converges, then so does (2), and to the same sum as (1). If the series (2) converges (in this case (1) may diverge), then the series (1) is called Euler summable.
  
If (1) converges, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e0366204.png" />, if the sequences
+
If (1) converges, if $  a _ {n} > 0 $,  
 +
if the sequences
  
<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/e/e036/e036620/e0366205.png" /></td> </tr></table>
+
$$
 +
\Delta  ^ {k} a _ {n}  = \sum _ { l= } 0 ^ { k }
 +
( - 1)  ^ {l} \left ( \begin{array}{c}
 +
k \\
 +
l
 +
\end{array}
 +
\right ) a _ {n+} l ,\ \
 +
k = 0 , 1 \dots
 +
$$
  
 
are monotone, and if
 
are monotone, 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/e/e036/e036620/e0366206.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{a _ {n+} 1 }{a _ {n} }
 +
  \geq  q  >
 +
\frac{1}{2}
 +
,
 +
$$
  
 
then the series (2) converges more rapidly than (1) (see [[Convergence, types of|Convergence, types of]]).
 
then the series (2) converges more rapidly than (1) (see [[Convergence, types of|Convergence, types of]]).
Line 27: Line 67:
 
Euler's transformation is the integral transformation
 
Euler's transformation is the integral transformation
  
<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/e/e036/e036620/e0366207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
w ( z)  = \int\limits _ { C } ( z - t )  ^  \alpha  v ( t) dt ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e0366208.png" /> is a contour in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e0366209.png" />-plane. It was proposed by L. Euler (1769).
+
where $  C $
 +
is a contour in the complex $  t $-
 +
plane. It was proposed by L. Euler (1769).
  
 
The Euler transformation is applied to linear ordinary differential equations of the form
 
The Euler transformation is applied to linear ordinary differential equations of the form
  
<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/e/e036/e036620/e03662010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
L w  = \sum _ { j= } 0 ^ { n }  (- 1)  ^ {j} w  ^ {(} j)
 +
\sum _ { k= } 0 ^ { n- }  j \left ( \begin{array}{c}
 +
n + \beta - j - 1 \\
 +
 
 +
n - k - j
 +
\end{array}
 +
\right ) Q _ {j}  ^ {(} n- k- j) ( z) =  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662011.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662013.png" /> is a constant. Any linear equation of the form
+
where $  Q _ {j} ( z) $
 +
is a polynomial of degree $  \leq  n - j $
 +
and $  \beta $
 +
is a constant. Any linear equation of the form
  
<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/e/e036/e036620/e03662014.png" /></td> </tr></table>
+
$$
 +
P _ {n} ( z) w  ^ {(} n) + P _ {n-} 1 ( z)
 +
w  ^ {(} n- 1) + \dots + P _ {0} ( z) w  = 0 ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662015.png" /> are polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662016.png" /> and the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662017.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662018.png" />, can be written in the form (2). The equation
+
where the $  P _ {j} ( z) $
 +
are polynomials of degree $  \leq  j $
 +
and the degree of $  P _ {n} ( z) $
 +
is $  n $,  
 +
can be written in the form (2). The equation
  
<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/e/e036/e036620/e03662019.png" /></td> </tr></table>
+
$$
 +
M v  \equiv  \sum _ { j= } 0 ^ { n }  (- 1)  ^ {j}
 +
( Q _ {n-} j ( z) v )  ^ {(} j)  = 0
 +
$$
  
is called the Euler transform of (2). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662020.png" /> is defined by (1) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662021.png" />, then
+
is called the Euler transform of (2). If $  w ( z) $
 +
is defined by (1) and $  \alpha = \beta + n - 1 $,  
 +
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/e/e036/e036620/e03662022.png" /></td> </tr></table>
+
$$
 +
L w  = \int\limits _ { C } ( z - t )  ^  \alpha  M ( v)  dt ,
 +
$$
  
provided that the integrated term arising from integration by parts vanishes. From this it follows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662024.png" /> is a solution of (2).
+
provided that the integrated term arising from integration by parts vanishes. From this it follows that if $  M ( v) = 0 $,  
 +
then $  w ( z) $
 +
is a solution of (2).
  
The Euler transformation makes it possible to reduce the order of (2) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662027.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662029.png" /> equation (2) can be integrated (see [[Pochhammer equation|Pochhammer equation]]).
+
The Euler transformation makes it possible to reduce the order of (2) if $  Q _ {n-} j ( z) \equiv 0 $
 +
for $  j > q $,  
 +
$  q < n $.  
 +
For $  q = 0 $
 +
and $  q= 1 $
 +
equation (2) can be integrated (see [[Pochhammer equation|Pochhammer equation]]).
  
 
====References====
 
====References====
Line 58: Line 134:
 
The Euler transform of the first kind is the integral transform
 
The Euler transform of the first kind is the integral transform
  
<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/e/e036/e036620/e03662030.png" /></td> </tr></table>
+
$$
 +
I  ^  \mu  f ( x)  = \mathfrak K _  \mu  \{ f ( t) ; x \}  = \
 +
 
 +
\frac{1}{\Gamma ( \mu ) }
 +
\int\limits _ { 0 } ^ { x }  f ( t)
 +
( x - t ) ^ {\mu - 1 }  dt ,
 +
$$
 +
 
 +
where  $  \mu $
 +
and  $  x $
 +
are complex variables and the path of integration is the segment  $  t = x \tau $,
 +
0 < \tau < 1 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662032.png" /> are complex variables and the path of integration is the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662034.png" />.
+
The Euler transform of the first kind is also called the fractional Riemann–Liouville integral of order  $  \mu $.  
 +
(Sometimes the name of Riemann–Liouville integral is given to
  
The Euler transform of the first kind is also called the fractional Riemann–Liouville integral of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662036.png" />. (Sometimes the name of Riemann–Liouville integral is given to
+
$$
  
<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/e/e036/e036620/e03662037.png" /></td> </tr></table>
+
\frac{1}{\Gamma ( \mu ) }
 +
\int\limits _ { x } ^ { a }  f ( t)
 +
( t - x ) ^ {\mu - 1 }  dt  = \mathfrak K _  \mu  \{ f ( a - t ) ; \
 +
a - x \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662038.png" /> is a complex number.)
+
where $  a $
 +
is a complex number.)
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662040.png" /> satisfy certain conditions, then
+
If $  f $
 +
and $  g $
 +
satisfy certain conditions, 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/e/e036/e036620/e03662041.png" /></td> </tr></table>
+
$$
 +
I  ^ {0} f ( x)  = f ( x) ,
 +
$$
  
<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/e/e036/e036620/e03662042.png" /></td> </tr></table>
+
$$
 +
I  ^  \mu  [ \alpha f ( x) + \beta g ( x) ]  = \alpha I  ^  \mu  f ( x) + \beta I  ^  \mu  g ( x) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662044.png" /> are complex constants and
+
where $  \alpha $
 +
and $  \beta $
 +
are complex constants and
  
<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/e/e036/e036620/e03662045.png" /></td> </tr></table>
+
$$
 +
I  ^  \mu  [ I  ^  \nu  f ( x) ]  = I ^ {\mu + \nu } f ( x) ,
 +
$$
  
<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/e/e036/e036620/e03662046.png" /></td> </tr></table>
+
$$
 +
I  ^ {n} f ( x)  = \int\limits _ { 0 } ^ { x }  dt _ {1} \dots
 +
\int\limits _ { 0 } ^ { {t _ n-} 2 } d t _ {n-} 1 \int\limits _ { 0 } ^ { {t } _ {n-} 1 } f ( t _ {n} )  d t _ {n} ,
 +
$$
  
<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/e/e036/e036620/e03662047.png" /></td> </tr></table>
+
$$
 +
I  ^ {-} n f ( x)  =
 +
\frac{d  ^ {n} }{d x  ^ {n} }
 +
f ( x) ,\  n = 1 , 2 ,\dots .
 +
$$
  
 
The Euler transform of the second kind is the integral transform
 
The Euler transform of the second kind is the integral transform
  
<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/e/e036/e036620/e03662048.png" /></td> </tr></table>
+
$$
 +
K  ^  \mu  f ( x)  = \mathfrak M _  \mu  \{ f ( t) ; x \}  = \
 +
 
 +
\frac{1}{\Gamma ( \mu ) }
 +
\int\limits _ { x } ^  \infty  f ( t) ( t - x ) ^ {\mu - 1 }  d t ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662050.png" /> are complex variables and the path of integration is the ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662052.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662054.png" />. Under certain conditions,
+
where $  \mu $
 +
and $  x $
 +
are complex variables and the path of integration is the ray $  t = x \tau $,  
 +
$  \tau > 1 $,  
 +
or $  t = x + \tau $,
 +
$  \tau > 0 $.  
 +
Under certain conditions,
  
<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/e/e036/e036620/e03662055.png" /></td> </tr></table>
+
$$
 +
K  ^ {0} f ( x)  = f ( x) ,
 +
$$
  
<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/e/e036/e036620/e03662056.png" /></td> </tr></table>
+
$$
 +
K  ^  \mu  [ \alpha f ( x) + \beta g ( x) ]  = \alpha K  ^  \mu  f ( x) + \beta K  ^  \mu  ( x) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662058.png" /> are complex constants and
+
where $  \alpha $
 +
and $  \beta $
 +
are complex constants and
  
<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/e/e036/e036620/e03662059.png" /></td> </tr></table>
+
$$
 +
K  ^  \mu  [ K  ^  \nu  f ( x) ]  = K ^ {\mu + \nu } f ( x) ,
 +
$$
  
<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/e/e036/e036620/e03662060.png" /></td> </tr></table>
+
$$
 +
K  ^ {n} f ( x)  = \int\limits _ { x } ^  \infty  d t _ {1} \dots \int\limits _ {t _ {n-} 2 } ^  \infty 
 +
d t _ {n-} 1 \int\limits _ {t _ {n-} 1 } ^  \infty  f ( t _ {n} )  d t _ {n} ,
 +
$$
  
<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/e/e036/e036620/e03662061.png" /></td> </tr></table>
+
$$
 +
K  ^ {-} n f ( x)  =
 +
\frac{d  ^ {n} }{d x  ^ {n} }
 +
f ( x) ,\  n = 1 , 2 ,\dots .
 +
$$
  
The Euler transform of the second kind is sometimes called the fractional Weyl integral of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036620/e03662063.png" />.
+
The Euler transform of the second kind is sometimes called the fractional Weyl integral of order $  \mu $.
  
 
The above transforms have also been introduced for generalized functions.
 
The above transforms have also been introduced for generalized functions.

Revision as of 19:38, 5 June 2020


The Euler transformation of series. Given a series

$$ \tag{1 } \sum _ { n= } 0 ^ \infty (- 1) ^ {n} a _ {n} , $$

the series

$$ \tag{2 } \sum _ { n= } 0 ^ \infty \frac{\Delta ^ {n} a _ {0} }{2 ^ {n+} 1 } $$

is said to be obtained from (1) by means of the Euler transformation. Here

$$ \Delta ^ {n} a _ {0} = \sum _ { k= } 0 ^ { n } (- 1) ^ {k} \left ( \begin{array}{c} n \\ k \end{array} \right ) a _ {k} . $$

If the series (1) converges, then so does (2), and to the same sum as (1). If the series (2) converges (in this case (1) may diverge), then the series (1) is called Euler summable.

If (1) converges, if $ a _ {n} > 0 $, if the sequences

$$ \Delta ^ {k} a _ {n} = \sum _ { l= } 0 ^ { k } ( - 1) ^ {l} \left ( \begin{array}{c} k \\ l \end{array} \right ) a _ {n+} l ,\ \ k = 0 , 1 \dots $$

are monotone, and if

$$ \frac{a _ {n+} 1 }{a _ {n} } \geq q > \frac{1}{2} , $$

then the series (2) converges more rapidly than (1) (see Convergence, types of).

L.D. Kudryavtsev

Euler's transformation is the integral transformation

$$ \tag{1 } w ( z) = \int\limits _ { C } ( z - t ) ^ \alpha v ( t) dt , $$

where $ C $ is a contour in the complex $ t $- plane. It was proposed by L. Euler (1769).

The Euler transformation is applied to linear ordinary differential equations of the form

$$ \tag{2 } L w = \sum _ { j= } 0 ^ { n } (- 1) ^ {j} w ^ {(} j) \sum _ { k= } 0 ^ { n- } j \left ( \begin{array}{c} n + \beta - j - 1 \\ n - k - j \end{array} \right ) Q _ {j} ^ {(} n- k- j) ( z) = 0 , $$

where $ Q _ {j} ( z) $ is a polynomial of degree $ \leq n - j $ and $ \beta $ is a constant. Any linear equation of the form

$$ P _ {n} ( z) w ^ {(} n) + P _ {n-} 1 ( z) w ^ {(} n- 1) + \dots + P _ {0} ( z) w = 0 , $$

where the $ P _ {j} ( z) $ are polynomials of degree $ \leq j $ and the degree of $ P _ {n} ( z) $ is $ n $, can be written in the form (2). The equation

$$ M v \equiv \sum _ { j= } 0 ^ { n } (- 1) ^ {j} ( Q _ {n-} j ( z) v ) ^ {(} j) = 0 $$

is called the Euler transform of (2). If $ w ( z) $ is defined by (1) and $ \alpha = \beta + n - 1 $, then

$$ L w = \int\limits _ { C } ( z - t ) ^ \alpha M ( v) dt , $$

provided that the integrated term arising from integration by parts vanishes. From this it follows that if $ M ( v) = 0 $, then $ w ( z) $ is a solution of (2).

The Euler transformation makes it possible to reduce the order of (2) if $ Q _ {n-} j ( z) \equiv 0 $ for $ j > q $, $ q < n $. For $ q = 0 $ and $ q= 1 $ equation (2) can be integrated (see Pochhammer equation).

References

[1] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956)
[2] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Teubner (1943)

M.V. Fedoryuk

The Euler transform of the first kind is the integral transform

$$ I ^ \mu f ( x) = \mathfrak K _ \mu \{ f ( t) ; x \} = \ \frac{1}{\Gamma ( \mu ) } \int\limits _ { 0 } ^ { x } f ( t) ( x - t ) ^ {\mu - 1 } dt , $$

where $ \mu $ and $ x $ are complex variables and the path of integration is the segment $ t = x \tau $, $ 0 < \tau < 1 $.

The Euler transform of the first kind is also called the fractional Riemann–Liouville integral of order $ \mu $. (Sometimes the name of Riemann–Liouville integral is given to

$$ \frac{1}{\Gamma ( \mu ) } \int\limits _ { x } ^ { a } f ( t) ( t - x ) ^ {\mu - 1 } dt = \mathfrak K _ \mu \{ f ( a - t ) ; \ a - x \} , $$

where $ a $ is a complex number.)

If $ f $ and $ g $ satisfy certain conditions, then

$$ I ^ {0} f ( x) = f ( x) , $$

$$ I ^ \mu [ \alpha f ( x) + \beta g ( x) ] = \alpha I ^ \mu f ( x) + \beta I ^ \mu g ( x) , $$

where $ \alpha $ and $ \beta $ are complex constants and

$$ I ^ \mu [ I ^ \nu f ( x) ] = I ^ {\mu + \nu } f ( x) , $$

$$ I ^ {n} f ( x) = \int\limits _ { 0 } ^ { x } dt _ {1} \dots \int\limits _ { 0 } ^ { {t _ n-} 2 } d t _ {n-} 1 \int\limits _ { 0 } ^ { {t } _ {n-} 1 } f ( t _ {n} ) d t _ {n} , $$

$$ I ^ {-} n f ( x) = \frac{d ^ {n} }{d x ^ {n} } f ( x) ,\ n = 1 , 2 ,\dots . $$

The Euler transform of the second kind is the integral transform

$$ K ^ \mu f ( x) = \mathfrak M _ \mu \{ f ( t) ; x \} = \ \frac{1}{\Gamma ( \mu ) } \int\limits _ { x } ^ \infty f ( t) ( t - x ) ^ {\mu - 1 } d t , $$

where $ \mu $ and $ x $ are complex variables and the path of integration is the ray $ t = x \tau $, $ \tau > 1 $, or $ t = x + \tau $, $ \tau > 0 $. Under certain conditions,

$$ K ^ {0} f ( x) = f ( x) , $$

$$ K ^ \mu [ \alpha f ( x) + \beta g ( x) ] = \alpha K ^ \mu f ( x) + \beta K ^ \mu ( x) , $$

where $ \alpha $ and $ \beta $ are complex constants and

$$ K ^ \mu [ K ^ \nu f ( x) ] = K ^ {\mu + \nu } f ( x) , $$

$$ K ^ {n} f ( x) = \int\limits _ { x } ^ \infty d t _ {1} \dots \int\limits _ {t _ {n-} 2 } ^ \infty d t _ {n-} 1 \int\limits _ {t _ {n-} 1 } ^ \infty f ( t _ {n} ) d t _ {n} , $$

$$ K ^ {-} n f ( x) = \frac{d ^ {n} }{d x ^ {n} } f ( x) ,\ n = 1 , 2 ,\dots . $$

The Euler transform of the second kind is sometimes called the fractional Weyl integral of order $ \mu $.

The above transforms have also been introduced for generalized functions.

References

[1] Y.A. Brychkov, A.P. Prudnikov, "Integral transformations of generalized functions" , Gordon & Breach (1988) (Translated from Russian)

Yu.A. BrychkovA.P. Prudnikov

Comments

See also Fractional integration and differentiation.

References

[a1] A. Erdélyi, W. Magnus, F. Oberhetinger, F.G. Tricomi, "Tables of integral transforms" , II , McGraw-Hill (1954) pp. Chapt. 13
[a2] A.C. McBride, "Fractional calculus and integral transforms of generalized functions" , Pitman (1979)
How to Cite This Entry:
Euler transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euler_transformation&oldid=46861
This article was adapted from an original article by L.D. Kudryavtsev, M.V. Fedoryuk, Yu.A. Brychkov, A.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article