Difference between revisions of "Whittaker transform"
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The integral transform | The integral transform | ||
| − | $$F(x)=\int\limits_0^\infty(2xt)^{-1/4}W_{\lambda,\mu}(2xt)f(t)dt,$$ | + | $$F(x)=\int\limits_0^\infty(2xt)^{-1/4}W_{\lambda,\mu}(2xt)f(t)\,dt,$$ |
| − | where $W_{\lambda,\mu}(z)$ is the Whittaker function (cf. [[ | + | 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> |
| − | + | <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> | |
| − | + | <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> | |
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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=32643
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