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Difference between revisions of "Meijer transform"

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$$  
 
$$  
F( x)  =  \int\limits _ { 0 } ^  \infty  e  ^ {-} xt/2 ( xt) ^ {- \mu - 1/2 } W _ {\mu +
+
F( x)  =  \int\limits _ { 0 } ^  \infty  e  ^ {-xt/2} ( xt) ^ {- \mu - 1/2 } W _ {\mu +
 
1/2, \nu }  ( xt) f( t)  dt,
 
1/2, \nu }  ( xt) f( t)  dt,
 
$$
 
$$
  
 
where  $  W _ {\mu , \nu }  ( x) $
 
where  $  W _ {\mu , \nu }  ( x) $
is the Whittaker function (cf. [[Whittaker functions|Whittaker functions]]). The corresponding inversion formula is
+
is the Whittaker function (cf. [[Whittaker functions]]). The corresponding inversion formula is
  
 
$$  
 
$$  
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$$  
 
$$  
 
\times  
 
\times  
\int\limits _ {\beta - i \lambda } ^ {  \beta  + i \lambda } e
+
\int\limits _ {\beta - i \lambda } ^ {  \beta  + i \lambda } e^ {xt/2} ( xt) ^ {\mu - 1/2 } W _ {\mu - 1/2, \nu }  ( xt) F( x)  dx.
^ {xt/2} ( xt) ^ {\mu - 1/2 } W _ {\mu - 1/2, \nu }  ( xt) F( x)  dx.
 
 
$$
 
$$
  
 
For  $  \mu = \pm  \nu $
 
For  $  \mu = \pm  \nu $
the Meijer transform becomes the [[Laplace transform|Laplace transform]]; for  $  \mu = - 1/2 $
+
the Meijer transform becomes the [[Laplace transform]]; for  $  \mu = - 1/2 $
 
it becomes the  $  K _  \nu  $-
 
it becomes the  $  K _  \nu  $-
 
transform
 
transform
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where  $  K _  \nu  ( x) $
 
where  $  K _  \nu  ( x) $
is the [[Macdonald function|Macdonald function]].
+
is the [[Macdonald function]].
  
 
The Varma transform
 
The Varma transform

Latest revision as of 13:26, 13 January 2024


The integral transform

$$ F( x) = \int\limits _ { 0 } ^ \infty e ^ {-xt/2} ( xt) ^ {- \mu - 1/2 } W _ {\mu + 1/2, \nu } ( xt) f( t) dt, $$

where $ W _ {\mu , \nu } ( x) $ is the Whittaker function (cf. Whittaker functions). The corresponding inversion formula is

$$ f( t) = \lim\limits _ {\lambda \rightarrow + \infty } \frac{1}{2 \pi i } \frac{\Gamma ( 1- \mu + \nu ) }{\Gamma ( 1+ 2 \nu ) } \times $$

$$ \times \int\limits _ {\beta - i \lambda } ^ { \beta + i \lambda } e^ {xt/2} ( xt) ^ {\mu - 1/2 } W _ {\mu - 1/2, \nu } ( xt) F( x) dx. $$

For $ \mu = \pm \nu $ the Meijer transform becomes the Laplace transform; for $ \mu = - 1/2 $ it becomes the $ K _ \nu $- transform

$$ F( x) = \frac{1}{\sqrt \pi } \int\limits _ { 0 } ^ \infty e ^ {-} xt/2 ( xt) ^ {1/2} K _ \nu \left ( \frac{xt}{2} \right ) f( t) dt, $$

where $ K _ \nu ( x) $ is the Macdonald function.

The Varma transform

$$ F( x) = \int\limits _ { 0 } ^ \infty ( xt) ^ {\nu - 1/2 } e ^ {-} xt/2 W _ {\mu , \nu } ( xt) f( t) dt $$

reduces to a Meijer transform.

The Meijer $ K $- transform (or the Meijer–Bessel transform) is the integral transform

$$ F( x) = \sqrt { \frac{2} \pi } \int\limits _ { 0 } ^ \infty K _ \nu ( xt) \sqrt xt f( t) dt. $$

If the function $ f $ is locally integrable on $ ( 0, \infty ) $, has bounded variation in a neighbourhood of the point $ t = t _ {0} > 0 $, and if the integral

$$ \int\limits _ { 0 } ^ \infty e ^ {- \beta t } | f( t) | dt,\ \ \beta > \alpha \geq 0, $$

converges, then the following inversion formula is valid:

$$ \frac{f( t _ {0} + 0) + f( t _ {0} - 0) }{2\ } = $$

$$ = \ \lim\limits _ {\lambda \rightarrow \infty } \frac{1}{i \sqrt {2 \pi } } \int\limits _ {\beta - i \lambda } ^ { \beta + i \lambda } I _ \nu ( t _ {0} x)( t _ {0} x) ^ {1/2} F( x) dx. $$

For $ \nu = \pm 1/2 $ the Meijer $ K $- transform turns into the Laplace transform.

The Meijer transform and Meijer $ K $- transform were introduced by C.S. Meijer in [1] and, respectively, .

References

[1] C.S. Meijer, "Eine neue Erweiterung der Laplace Transformation I" Proc. Koninkl. Ned. Akad. Wet. , 44 (1941) pp. 727–737
[2a] C.S. Meijer, "Ueber eine neue Erweiterung der Laplace Transformation I" Proc. Koninkl. Ned. Akad. Wet. , 43 (1940) pp. 599–608
[2b] C.S. Meijer, "Ueber eine neue Erweiterung der Laplace Transformation II" Proc. Koninkl. Ned. Akad. Wet. , 43 (1940) pp. 702–711
[3] Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)
[4] V.A. Ditkin, A.P. Prudnikov, "Operational calculus" Progress in Math. , 1 (1968) pp. 1–75 Itogi Nauk. Mat. Anal. 1966 (1967) pp. 7–82
How to Cite This Entry:
Meijer transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Meijer_transform&oldid=55041
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