Anger function: Difference between revisions

Revision as of 08:49, 6 August 2010

Revision as of 00:36, 4 September 2011

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:<math>-\sin(\pi \nu)\mathbf{E}_\nu(z) = \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z)</math>

so in particular if &nu;ν is not an integer they can be expressed as linear combinations of each other. If &nu;ν is an integer then Anger functions '''J'''_&nu;ν are the same as Bessel functions ''J''_&nu;ν, and Weber functions can be expressed as finite linear combinations of [[Struve function]]s.

==Differential equations==

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the Anger functions satisfy the equation

:<math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = (z-\nu)\sin(\pi z)/\pi</math>

and the Weber functions satisfy the equation

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==References==

*{{AS ref|12|498}}

*C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig , 5 (1855) pp. &nbsp;1–29

*{{dlmf|id=11.10|title=Anger-Weber Functions|first=R. B. |last=Paris}}

*{{springer|id=A/a012490|title=Anger function|first=A.P.|last= Prudnikov}}

*{{springer|id=W/w097320|title=Weber function|first=A.P.|last= Prudnikov}}

*[[G.N. Watson]], "A treatise on the theory of Bessel functions" , 1–2 , Cambridge Univ. Press (1952)

*H.F. Weber, Zurich Vierteljahresschrift , 24 (1879) pp. &nbsp;33–76

[[Category:Special functions]]