Anger function: Difference between revisions

Revision as of 11:48, 17 December 2019

Revision as of 05:30, 20 December 2019

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

==Power series expansion==

*The '''Anger function''' has the power series expansion<ref name=DLMF>{{dlmf|id=11.10|title=Anger-Weber Functions|first=R. B. |last=Paris}}</ref>

:<math>\mathbf{J}_\nu(z)=\sum{k=0}^\infty\frac{\cos\frac{\pi\nu}{2}(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sum{k=0}^\infty\frac{\sin\frac{\pi\nu}{2}(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math>

While the '''Weber function''' has the power series expansion<ref name=DLMF/>

:<math>\mathbf{E}_\nu(z)=\sum{k=0}^\infty\frac{\sin\frac{\pi\nu}{2}(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}-\sum{k=0}^\infty\frac{\cos\frac{\pi\nu}{2}(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</math>

==Differential equations==

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*{{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|authorlink=Anatolii Platonovich 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

{{Reflist}}

[[Category:Special functions]]