alt=Plot of the Anger function J v(z) with n=2 from -2-2i to 2+2i|thumb|Plot of the Anger function {{math|'''J'''<sub>''ν''</sub>(''z'')}} with {{math|1=''n'' = 2}} from {{math|−2 − 2i}} to {{math|2 + 2''i''}} 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> with complex parameter <math>\nu</math> and complex variable {{tmath| \textit{z} }}.<ref name="EOM_Anger">{{springer |id=A/a012490 |title=Anger function |first=A.P. |last=Prudnikov |authorlink=Anatolii Platonovich Prudnikov }}</ref> It is closely related to the Bessel functions.
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 functions of the second kind.
== Relation between Weber and Anger functions ==
alt=Plot of the Weber function E v(z) with n=2 from -2-2i to 2+2i|thumb|Plot of the Weber function {{math|'''E'''<sub>''ν''</sub>(''z'')}} with {{math|1=''n'' = 2}} from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}} The Anger and Weber functions are related by : <math> \begin{align} \sin(\pi \nu)\mathbf{J}_\nu(z) &= \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z), \\ -\sin(\pi \nu)\mathbf{E}_\nu(z) &= \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z), \end{align} </math> so in particular if {{math|''ν''}} is not an integer they can be expressed as linear combinations of each other. If {{math|''ν''}} is an integer then Anger functions {{math|'''J'''<sub>''ν''</sub>}} are the same as Bessel functions {{math|''J''<sub>''ν''</sub>}}, and Weber functions can be expressed as finite linear combinations of Struve functions.
== 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)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-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)=\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}-\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-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 == The Anger and Weber functions are solutions of inhomogeneous 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<ref name=DLMF/> : <math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = \frac{(z-\nu)\sin(\pi \nu)}{\pi} ,</math> and the Weber functions satisfy the equation<ref name=DLMF/> : <math>z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = -\frac{z+\nu+(z-\nu)\cos(\pi \nu)}{\pi}.</math>
== Recurrence relations == The Anger function satisfies this inhomogeneous form of recurrence relation<ref name=DLMF/> : <math>z\mathbf{J}_{\nu-1}(z)+z\mathbf{J}_{\nu+1}(z)=2\nu\mathbf{J}_\nu(z)-\frac{2\sin\pi\nu}{\pi}.</math>
While the Weber function satisfies this inhomogeneous form of recurrence relation<ref name=DLMF/> : <math>z\mathbf{E}_{\nu-1}(z)+z\mathbf{E}_{\nu+1}(z)=2\nu\mathbf{E}_\nu(z)-\frac{2(1-\cos\pi\nu)}{\pi}.</math>
== Delay differential equations == The Anger and Weber functions satisfy these homogeneous forms of delay differential equations<ref name=DLMF/> : <math>\mathbf{J}_{\nu-1}(z)-\mathbf{J}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z),</math> : <math>\mathbf{E}_{\nu-1}(z)-\mathbf{E}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z).</math>
The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations<ref name=DLMF/> : <math>z\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z)\pm\nu\mathbf{J}_\nu(z)=\pm z\mathbf{J}_{\nu\mp1}(z)\pm\frac{\sin\pi\nu}{\pi},</math> : <math>z\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z)\pm\nu\mathbf{E}_\nu(z)=\pm z\mathbf{E}_{\nu\mp1}(z)\pm\frac{1-\cos\pi\nu}{\pi}.</math>
== References == {{reflist}} {{refbegin}} * {{AS ref|12|498}} * C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29 * {{springer|id=W/w097320|title=Weber function|first=A.P.|last= Prudnikov}} * {{citation |first1=G.N. |last1=Watson |author-link=G.N. Watson |title=A treatise on the theory of Bessel functions |pages=1–2 |publisher=Cambridge Univ. Press |date=1952 }} * H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76 {{refend}}
Category:Special functions