Anger function: Difference between revisions

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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>

Which pointly extracts from the Schläfli’s integral representation of '''Bessel functions of the first kind''' <math>J_\nu(z)=\frac{1}{\pi}\int_0^\pi\cos(\nu\theta-z\sin\theta)\,d\theta-\frac{\sin\nu\pi}{\pi}\int_0^\infty e^{-z\sinh t-\nu t}\,dt.</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>

Which pointly extracts from the Schläfli’s integral representation of '''Bessel functions of the second kind''' <math>Y_\nu(z)=\frac{1}{\pi}\int_0^\pi\sin(\nu\theta-z\sin\theta)\,d\theta-\frac{1}{\pi}\int_0^\infty(e^{\nu t}+e^{-\nu t}\cos\nu\pi)e^{-z\sinh t}\,dt.</math>

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

==Relation between Weber and Anger functions==