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The [[Integral transform|integral transform]]
 
The [[Integral transform|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/m/m063/m063370/m0633701.png" /></td> </tr></table>
+
$$
 +
F( x)  = \int\limits _ { 0 } ^  \infty  e  ^ {-xt/2} ( xt) ^ {- \mu - 1/2 } W _ {\mu +
 +
1/2, \nu }  ( xt) f( t)  dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m0633702.png" /> is the Whittaker function (cf. [[Whittaker functions|Whittaker functions]]). The corresponding inversion formula is
+
where $  W _ {\mu , \nu }  ( x) $
 +
is the Whittaker function (cf. [[Whittaker functions]]). The corresponding 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/m/m063/m063370/m0633703.png" /></td> </tr></table>
+
$$
 +
f( t)  = \lim\limits _ {\lambda \rightarrow + \infty } 
 +
\frac{1}{2 \pi i }
 +
 +
\frac{\Gamma ( 1- \mu + \nu ) }{\Gamma ( 1+ 2 \nu ) }
 +
\times
 +
$$
  
<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/m/m063/m063370/m0633704.png" /></td> </tr></table>
+
$$
 +
\times
 +
\int\limits _ {\beta - i \lambda } ^ {  \beta  + i \lambda } e^ {xt/2} ( xt) ^ {\mu - 1/2 } W _ {\mu - 1/2, \nu }  ( xt) F( x)  dx.
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m0633705.png" /> the Meijer transform becomes the [[Laplace transform|Laplace transform]]; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m0633706.png" /> it becomes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m0633708.png" />-transform
+
For $  \mu = \pm  \nu $
 +
the Meijer transform becomes the [[Laplace transform]]; for $  \mu = - 1/2 $
 +
it becomes the $  K _  \nu  $-
 +
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/m/m063/m063370/m0633709.png" /></td> </tr></table>
+
$$
 +
F( x)  =
 +
\frac{1}{\sqrt \pi }
 +
\int\limits _ { 0 } ^  \infty  e  ^ {-} xt/2 ( xt)  ^ {1/2} K _  \nu  \left (
 +
\frac{xt}{2}
 +
\right ) f( t)  dt,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337010.png" /> is the [[Macdonald function|Macdonald function]].
+
where $  K _  \nu  ( x) $
 +
is the [[Macdonald function]].
  
 
The Varma transform
 
The Varma 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/m/m063/m063370/m06337011.png" /></td> </tr></table>
+
$$
 +
F( x)  = \int\limits _ { 0 } ^  \infty  ( xt) ^ {\nu - 1/2 } e  ^ {-} xt/2 W _ {\mu , \nu }
 +
( xt) f( t)  dt
 +
$$
  
 
reduces to a Meijer transform.
 
reduces to a Meijer transform.
  
The Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337013.png" />-transform (or the Meijer–Bessel transform) is the integral transform
+
The Meijer $  K $-
 +
transform (or the Meijer–Bessel transform) 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/m/m063/m063370/m06337014.png" /></td> </tr></table>
+
$$
 +
F( x)  = \sqrt {
 +
\frac{2} \pi
 +
} \int\limits _ { 0 } ^  \infty  K _  \nu  ( xt) \sqrt xt f( t)  dt.
 +
$$
  
If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337015.png" /> is locally integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337016.png" />, has bounded variation in a neighbourhood of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337017.png" />, and if the integral
+
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
  
<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/m/m063/m063370/m06337018.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty  e ^ {- \beta t } | f( t) |  dt,\ \
 +
\beta > \alpha \geq  0,
 +
$$
  
 
converges, then the following inversion formula is valid:
 
converges, then the following inversion formula is valid:
  
<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/m/m063/m063370/m06337019.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{f( t _ {0} + 0) + f( t _ {0} - 0) }{2\ }
 +
=
 +
$$
  
<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/m/m063/m063370/m06337020.png" /></td> </tr></table>
+
$$
 +
= \
 +
\lim\limits _ {\lambda \rightarrow \infty } 
 +
\frac{1}{i \sqrt {2 \pi } }
 +
\int\limits _ {\beta - i \lambda
 +
} ^ {  \beta  + i \lambda } I _  \nu  ( t _ {0} x)( t _ {0} x)  ^ {1/2} F( x)  dx.
 +
$$
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337021.png" /> the Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337022.png" />-transform turns into the Laplace transform.
+
For $  \nu = \pm  1/2 $
 +
the Meijer $  K $-
 +
transform turns into the Laplace transform.
  
The Meijer transform and Meijer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063370/m06337023.png" />-transform were introduced by C.S. Meijer in [[#References|[1]]] and, respectively, .
+
The Meijer transform and Meijer $  K $-
 +
transform were introduced by C.S. Meijer in [[#References|[1]]] and, respectively, .
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.S. Meijer,  "Eine neue Erweiterung der Laplace Transformation I"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''44'''  (1941)  pp. 727–737</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  C.S. Meijer,  "Ueber eine neue Erweiterung der Laplace Transformation I"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''43'''  (1940)  pp. 599–608</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  C.S. Meijer,  "Ueber eine neue Erweiterung der Laplace Transformation II"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''43'''  (1940)  pp. 702–711</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Y.A. Brychkov,  A.P. Prudnikov,  "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.S. Meijer,  "Eine neue Erweiterung der Laplace Transformation I"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''44'''  (1941)  pp. 727–737</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  C.S. Meijer,  "Ueber eine neue Erweiterung der Laplace Transformation I"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''43'''  (1940)  pp. 599–608</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  C.S. Meijer,  "Ueber eine neue Erweiterung der Laplace Transformation II"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''43'''  (1940)  pp. 702–711</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Y.A. Brychkov,  A.P. Prudnikov,  "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  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</TD></TR></table>

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=12567
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