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Difference between revisions of "Laguerre transform"

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$$  
 
$$  
 
f ( n)  =  T \{ F ( x) \}  = \  
 
f ( n)  =  T \{ F ( x) \}  = \  
\int\limits _ { 0 } ^  \infty  e  ^ {-} x L _ {n} ( x) F ( x)  d x ,\ \  
+
\int\limits _ { 0 } ^  \infty  e  ^ {- x} L _ {n} ( x) F ( x)  d x ,\ \  
 
n = 0, 1 \dots
 
n = 0, 1 \dots
 
$$
 
$$
  
 
where  $  L _ {n} ( x) $
 
where  $  L _ {n} ( x) $
is the Laguerre polynomial (cf. [[Laguerre polynomials|Laguerre polynomials]]) of degree  $  n $.  
+
is the Laguerre polynomial (cf. [[Laguerre polynomials]]) of degree  $  n $.  
 
The inversion formula is
 
The inversion formula is
  
 
$$  
 
$$  
T  ^ {-} 1 \{ f ( n) \}  =  F ( x)  = \  
+
T  ^ {-1} \{ f ( n) \}  =  F ( x)  = \  
\sum _ { n= } 0 ^  \infty  f ( n) L _ {n} ( x) ,\ \  
+
\sum _ { n= 0} ^  \infty  f ( n) L _ {n} ( x) ,\ \  
 
0 < x < \infty ,
 
0 < x < \infty ,
 
$$
 
$$
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\frac{d F ( x) }{dx}
 
\frac{d F ( x) }{dx}
 
  \right \}  = \  
 
  \right \}  = \  
\sum _ { k= } 0 ^ { n }  f ( k) - F ( 0) ,\ \  
+
\sum _ { k=0} ^ { n }  f ( k) - F ( 0) ,\ \  
 
n = 0 , 1 \dots
 
n = 0 , 1 \dots
 
$$
 
$$
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\frac{d}{dx}
 
\frac{d}{dx}
  
\left [ x e  ^ {-} x
+
\left [ x e  ^ {-x}
  
 
\frac{d F ( x) }{dx}
 
\frac{d F ( x) }{dx}
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$$  
 
$$  
T  ^ {-} 1 \{ f ( n) g ( n) \} =
+
T  ^ {-1} \{ f ( n) g ( n) \} =
 
$$
 
$$
  
 
$$  
 
$$  
= \
+
= \frac{1} \pi  
 
+
  \int\limits _ { 0 } ^  \infty  e  ^ {-t} F ( t) \int\limits
\frac{1} \pi  
 
  \int\limits _ { 0 } ^  \infty  e  ^ {-} t F ( t) \int\limits
 
 
_ { 0 } ^  \pi  e ^ {\sqrt {xt } \cos  \theta
 
_ { 0 } ^  \pi  e ^ {\sqrt {xt } \cos  \theta
 
  } \cos ( \sqrt {xt } \sin  \theta ) \times
 
  } \cos ( \sqrt {xt } \sin  \theta ) \times
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$$  
 
$$  
 
= \  
 
= \  
\int\limits _ { 0 } ^  \infty  e  ^ {-} x x  ^  \alpha  L _ {n}  ^  \alpha  ( x) F ( x)  d x ,\  n = 0 , 1 \dots
+
\int\limits _ { 0 } ^  \infty  e  ^ {- x} x  ^  \alpha  L _ {n}  ^  \alpha  ( x) F ( x)  d x ,\  n = 0 , 1 \dots
 
$$
 
$$
  
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====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>

Latest revision as of 08:39, 6 January 2024


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