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The integral transform
 
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/l/l057/l057320/l0573201.png" /></td> </tr></table>
+
$$
 +
f ( n)  = T \{ F ( x) \}  = \
 +
\int\limits _ { 0 } ^  \infty  e  ^ {-} x L _ {n} ( x) F ( x)  d x ,\ \
 +
n = 0, 1 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573202.png" /> is the Laguerre polynomial (cf. [[Laguerre polynomials|Laguerre polynomials]]) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573203.png" />. The inversion formula is
+
where $  L _ {n} ( x) $
 +
is the Laguerre polynomial (cf. [[Laguerre polynomials|Laguerre polynomials]]) of degree $  n $.  
 +
The inversion formula is
  
<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/l/l057/l057320/l0573204.png" /></td> </tr></table>
+
$$
 +
T  ^ {-} 1 \{ f ( n) \}  = F ( x)  = \
 +
\sum _ { n= } 0 ^  \infty  f ( n) L _ {n} ( x) ,\ \
 +
0 < x < \infty ,
 +
$$
  
if the series converges. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573205.png" /> is continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573206.png" /> is piecewise continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573207.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l0573209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732010.png" />, then
+
if the series converges. If $  F $
 +
is continuous, $  F ^ { \prime } $
 +
is piecewise continuous on $  [ 0 , \infty ) $
 +
and $  F ( x) = O ( e  ^ {ax} ) $,  
 +
$  x \rightarrow \infty $,  
 +
$  a < 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/l/l057/l057320/l05732011.png" /></td> </tr></table>
+
$$
 +
T \left \{
 +
\frac{d F ( x) }{dx}
 +
\right \}  = \
 +
\sum _ { k= } 0 ^ { n }  f ( k) - F ( 0) ,\ \
 +
n = 0 , 1 \dots
 +
$$
  
<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/l/l057/l057320/l05732012.png" /></td> </tr></table>
+
$$
 +
T \left \{ x
 +
\frac{d F ( x) }{dx}
 +
\right \}  = - (
 +
n + 1 ) f ( n + 1 ) + n f ( n) ,\  n = 0 , 1 , \dots.
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732014.png" /> are continuous, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732015.png" /> is piecewise continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732019.png" />, then
+
If $  F $
 +
and $  F ^ { \prime } $
 +
are continuous, $  F ^ { \prime\prime } $
 +
is piecewise continuous on $  [ 0 , \infty ) $
 +
and $  | F ( x) | + | F ^ { \prime } ( x) | = O ( e  ^ {ax} ) $,  
 +
$  x \rightarrow \infty $,  
 +
$  a < 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/l/l057/l057320/l05732020.png" /></td> </tr></table>
+
$$
 +
T \left \{ e  ^ {x}
 +
\frac{d}{dx}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732021.png" /> is piecewise continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732025.png" />, then for
+
\left [ x e  ^ {-} 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/l/l057/l057320/l05732026.png" /></td> </tr></table>
+
\frac{d F ( x) }{dx}
 +
\right ] \right \}  = -
 +
n f ( n) ,\  n = 0 , 1 , .  . ..
 +
$$
  
<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/l/l057/l057320/l05732027.png" /></td> </tr></table>
+
If  $  F $
 +
is piecewise continuous on  $  [ 0 , \infty ) $
 +
and  $  F ( x) = O ( e  ^ {ax} ) $,
 +
$  x \rightarrow \infty $,
 +
$  a < 1 $,
 +
then for
  
and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732028.png" />,
+
$$
 +
G ( x)  = \int\limits _ { 0 } ^ { x }  F ( t)  d t ,
 +
$$
  
<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/l/l057/l057320/l05732029.png" /></td> </tr></table>
+
$$
 +
g ( n)  = T \left \{ \int\limits _ { 0 } ^ { x }  F ( t)  d t \right \}
 +
= f ( n) - f ( n - 1 ) ,\  n = 1 , 2 \dots
 +
$$
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732031.png" /> are piecewise continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732032.png" /> and that
+
and for  $  n = 0 $,
  
<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/l/l057/l057320/l05732033.png" /></td> </tr></table>
+
$$
 +
g ( 0)  = f ( 0) .
 +
$$
  
<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/l/l057/l057320/l05732034.png" /></td> </tr></table>
+
Suppose that  $  F $
 +
and  $  G $
 +
are piecewise continuous on  $  [ 0 , \infty ) $
 +
and that
 +
 
 +
$$
 +
| F ( x) | + | G ( x) |  = O ( e  ^ {ax} ) ,\ \
 +
x \rightarrow \infty ,\  a <  
 +
\frac{1}{2}
 +
,
 +
$$
 +
 
 +
$$
 +
T \{ F \}  = f ( n) ,\  T \{ G \}  = g ( n) .
 +
$$
  
 
Then
 
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/l/l057/l057320/l05732035.png" /></td> </tr></table>
+
$$
 +
T  ^ {-} 1 \{ f ( n) g ( n) \} =
 +
$$
 +
 
 +
$$
 +
= \
  
<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/l/l057/l057320/l05732036.png" /></td> </tr></table>
+
\frac{1} \pi
 +
\int\limits _ { 0 } ^  \infty  e  ^ {-} t F ( t) \int\limits
 +
_ { 0 } ^  \pi  e ^ {\sqrt {xt } \cos  \theta
 +
} \cos ( \sqrt {xt } \sin  \theta ) \times
 +
$$
  
<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/l/l057/l057320/l05732037.png" /></td> </tr></table>
+
$$
 +
\times
 +
G ( x + t - 2 \sqrt {xt } \cos  \theta )  d \theta  d t .
 +
$$
  
 
The generalized Laguerre transform is
 
The generalized Laguerre transform is
  
<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/l/l057/l057320/l05732038.png" /></td> </tr></table>
+
$$
 +
f _  \alpha  ( n)  = T _  \alpha  \{ F ( x) \} =
 +
$$
  
<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/l/l057/l057320/l05732039.png" /></td> </tr></table>
+
$$
 +
= \
 +
\int\limits _ { 0 } ^  \infty  e  ^ {-} x x  ^  \alpha  L _ {n}  ^  \alpha  ( x) F ( x)  d x ,\  n = 0 , 1 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057320/l05732040.png" /> is the generalized Laguerre polynomial (see [[#References|[4]]]).
+
where $  L _ {n}  ^  \alpha  ( x) $
 +
is the generalized Laguerre polynomial (see [[#References|[4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transformations" , Interscience  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. McCully,  "The Laguerre transform"  ''SIAM Rev.'' , '''2''' :  3  (1960)  pp. 185–191</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Debnath,  "On Laguerre transform"  ''Bull. Calcutta Math. Soc.'' , '''52''' :  2  (1960)  pp. 69–77</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.A. Brychkov,  A.P. Prudnikov,  "Operational calculus"  ''Progress in Math.'' , '''1'''  (1968)  pp. 1–74  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Zemanian,  "Generalized integral transformations" , Interscience  (1968)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. McCully,  "The Laguerre transform"  ''SIAM Rev.'' , '''2''' :  3  (1960)  pp. 185–191</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Debnath,  "On Laguerre transform"  ''Bull. Calcutta Math. Soc.'' , '''52''' :  2  (1960)  pp. 69–77</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.A. Brychkov,  A.P. Prudnikov,  "Operational calculus"  ''Progress in Math.'' , '''1'''  (1968)  pp. 1–74  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR></table>

Revision as of 22:15, 5 June 2020


The integral transform

$$ f ( n) = T \{ F ( x) \} = \ \int\limits _ { 0 } ^ \infty e ^ {-} x L _ {n} ( x) F ( x) d x ,\ \ n = 0, 1 \dots $$

where $ L _ {n} ( x) $ is the Laguerre polynomial (cf. Laguerre polynomials) of degree $ n $. The inversion formula is

$$ T ^ {-} 1 \{ f ( n) \} = F ( x) = \ \sum _ { n= } 0 ^ \infty f ( n) L _ {n} ( x) ,\ \ 0 < x < \infty , $$

if the series converges. If $ F $ is continuous, $ F ^ { \prime } $ is piecewise continuous on $ [ 0 , \infty ) $ and $ F ( x) = O ( e ^ {ax} ) $, $ x \rightarrow \infty $, $ a < 1 $, then

$$ T \left \{ \frac{d F ( x) }{dx} \right \} = \ \sum _ { k= } 0 ^ { n } f ( k) - F ( 0) ,\ \ n = 0 , 1 \dots $$

$$ T \left \{ x \frac{d F ( x) }{dx} \right \} = - ( n + 1 ) f ( n + 1 ) + n f ( n) ,\ n = 0 , 1 , \dots. $$

If $ F $ and $ F ^ { \prime } $ are continuous, $ F ^ { \prime\prime } $ is piecewise continuous on $ [ 0 , \infty ) $ and $ | F ( x) | + | F ^ { \prime } ( x) | = O ( e ^ {ax} ) $, $ x \rightarrow \infty $, $ a < 1 $, then

$$ T \left \{ e ^ {x} \frac{d}{dx} \left [ x e ^ {-} x \frac{d F ( x) }{dx} \right ] \right \} = - n f ( n) ,\ n = 0 , 1 , . . .. $$

If $ F $ is piecewise continuous on $ [ 0 , \infty ) $ and $ F ( x) = O ( e ^ {ax} ) $, $ x \rightarrow \infty $, $ a < 1 $, then for

$$ G ( x) = \int\limits _ { 0 } ^ { x } F ( t) d t , $$

$$ g ( n) = T \left \{ \int\limits _ { 0 } ^ { x } F ( t) d t \right \} = f ( n) - f ( n - 1 ) ,\ n = 1 , 2 \dots $$

and for $ n = 0 $,

$$ g ( 0) = f ( 0) . $$

Suppose that $ F $ and $ G $ are piecewise continuous on $ [ 0 , \infty ) $ and that

$$ | F ( x) | + | G ( x) | = O ( e ^ {ax} ) ,\ \ x \rightarrow \infty ,\ a < \frac{1}{2} , $$

$$ T \{ F \} = f ( n) ,\ T \{ G \} = g ( n) . $$

Then

$$ T ^ {-} 1 \{ f ( n) g ( n) \} = $$

$$ = \ \frac{1} \pi \int\limits _ { 0 } ^ \infty e ^ {-} t F ( t) \int\limits _ { 0 } ^ \pi e ^ {\sqrt {xt } \cos \theta } \cos ( \sqrt {xt } \sin \theta ) \times $$

$$ \times G ( x + t - 2 \sqrt {xt } \cos \theta ) d \theta d t . $$

The generalized Laguerre transform is

$$ f _ \alpha ( n) = T _ \alpha \{ F ( x) \} = $$

$$ = \ \int\limits _ { 0 } ^ \infty e ^ {-} x x ^ \alpha L _ {n} ^ \alpha ( x) F ( x) d x ,\ n = 0 , 1 \dots $$

where $ L _ {n} ^ \alpha ( x) $ is the generalized Laguerre polynomial (see [4]).

References

[1] A.G. Zemanian, "Generalized integral transformations" , Interscience (1968)
[2] J. McCully, "The Laguerre transform" SIAM Rev. , 2 : 3 (1960) pp. 185–191
[3] L. Debnath, "On Laguerre transform" Bull. Calcutta Math. Soc. , 52 : 2 (1960) pp. 69–77
[4] Yu.A. Brychkov, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–74 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82
How to Cite This Entry:
Laguerre transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Laguerre_transform&oldid=47567
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article