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Difference between revisions of "Whittaker transform"

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$$F(x)=\int\limits_0^\infty(2xt)^{-1/4}W_{\lambda,\mu}(2xt)f(t)\,dt,$$
 
$$F(x)=\int\limits_0^\infty(2xt)^{-1/4}W_{\lambda,\mu}(2xt)f(t)\,dt,$$
  
where $W_{\lambda,\mu}(z)$ is the Whittaker function (cf. [[Whittaker functions|Whittaker functions]]). For $\lambda=1/4$ and $\mu=\pm1/4$ the Whittaker transform goes over into the [[Laplace transform|Laplace transform]].
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where $W_{\lambda,\mu}(z)$ is the Whittaker function (cf. [[Whittaker functions]]). For $\lambda=1/4$ and $\mu=\pm1/4$ the Whittaker transform goes over into the [[Laplace transform]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.S. Meijer,  "Eine neue Erweiterung der Laplace-Transformation"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''44'''  (1941)  pp. 727–737</TD></TR></table>
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<table>
 
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<TR><TD valign="top">[1]</TD> <TD valign="top">  C.S. Meijer,  "Eine neue Erweiterung der Laplace-Transformation"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''44'''  (1941)  pp. 727–737</TD></TR>
 
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Doetsch,  "Handbuch der Laplace-Transformation" , '''III''' , Birkhäuser  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1927)</TD></TR>
 
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</table>
====Comments====
 
 
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Doetsch,  "Handbuch der Laplace-Transformation" , '''III''' , Birkhäuser  (1973)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.T. Whittaker,  G.N. Watson,  "A course of modern analysis" , Cambridge Univ. Press  (1927)</TD></TR></table>
 

Latest revision as of 14:09, 8 April 2023

The integral transform

$$F(x)=\int\limits_0^\infty(2xt)^{-1/4}W_{\lambda,\mu}(2xt)f(t)\,dt,$$

where $W_{\lambda,\mu}(z)$ is the Whittaker function (cf. Whittaker functions). For $\lambda=1/4$ and $\mu=\pm1/4$ the Whittaker transform goes over into the Laplace transform.

References

[1] C.S. Meijer, "Eine neue Erweiterung der Laplace-Transformation" Proc. Koninkl. Ned. Akad. Wet. , 44 (1941) pp. 727–737
[a1] G. Doetsch, "Handbuch der Laplace-Transformation" , III , Birkhäuser (1973)
[a2] E.T. Whittaker, G.N. Watson, "A course of modern analysis" , Cambridge Univ. Press (1927)
How to Cite This Entry:
Whittaker transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whittaker_transform&oldid=53673
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