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Difference between revisions of "Whittaker transform"

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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/w/w097/w097860/w0978601.png" /></td> </tr></table>
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$$F(x)=\int\limits_0^\infty(2xt)^{-1/4}W_{\lambda,\mu}(2xt)f(t)dt,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097860/w0978602.png" /> is the Whittaker function (cf. [[Whittaker functions|Whittaker functions]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097860/w0978603.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097860/w0978604.png" /> 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|Whittaker functions]]). For $\lambda=1/4$ and $\mu=\pm1/4$ the Whittaker transform goes over into the [[Laplace transform|Laplace transform]].
  
 
====References====
 
====References====

Revision as of 10:55, 1 August 2014

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


Comments

References

[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=32643
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