en>LilHelpa: Typo fixing and general fixes using AWB

2014-01-12T22:34:24Z

Typo fixing and general fixes using AWB

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In mathematics, the '''Anger function''', introduced by {{harvs|txt|authorlink=Carl Theodor Anger|first=C. T.|last=Anger|year=1855}}, is a function defined as

: <math>\mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta</math>

and is closely related to [[Bessel function]]s.

The '''Weber function''' (also known as '''Lommel-Weber function'''), introduced by {{harvs|txt|authorlink=Heinrich Friedrich Weber|first=H. F.|last=Weber|year=1879}}, is a closely related function defined by

: <math>\mathbf{E}_\nu(z)=\frac{1}{\pi} \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta</math>

and is closely related to [[Bessel function]]s of the second kind.

==Relation between Weber and Anger functions==

The Anger and Weber functions are related by

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

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

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'''<sub>ν</sub> are the same as Bessel functions ''J''<sub>ν</sub>, and Weber functions can be expressed as finite linear combinations of [[Struve function]]s.

==Differential equations==
The Anger and Weber functions are solutions of inhomogenous forms of Bessel's equation <math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0</math>. More precisely,
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

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

==References==
*{{AS ref|12|498}}
*C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 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. 33–76

[[Category:Special functions]]

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en>LilHelpa: Typo fixing and general fixes using AWB