# Anger function > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Anger_function > Markdown URL: https://mediated.wiki/source/Anger_function.md > Source: https://en.wikipedia.org/wiki/Anger_function > Source revision: 1327664251 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) 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'''''ν''(''z'')}} with {{math|1=''n'' = 2}} from {{math|−2 − 2i}} to {{math|2 + 2''i''}} In [mathematics](/source/mathematics), the '''Anger function''', introduced by {{harvs|txt |authorlink=Carl Theodor Anger |first=C. T. |last=Anger |year=1855 }}, is a function defined as : \mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta with complex parameter \nu and complex variable {{tmath| \textit{z} }}.{{springer |id=A/a012490 |title=Anger function |first=A.P. |last=Prudnikov |authorlink=Anatolii Platonovich Prudnikov }} It is closely related to the [Bessel function](/source/Bessel_function)s. The '''Weber function''' (also known as '''[Lommel](/source/Eugen_von_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 : \mathbf{E}_\nu(z)=\frac{1}{\pi} \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta and is closely related to [Bessel function](/source/Bessel_function)s 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'''''ν''(''z'')}} with {{math|1=''n'' = 2}} from {{math|−2 − 2''i''}} to {{math|2 + 2''i''}} The Anger and Weber functions are related by : \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} so in particular if {{math|''ν''}} is not an [integer](/source/integer) they can be expressed as linear combinations of each other. If {{math|''ν''}} is an integer then Anger functions {{math|'''J'''''ν''}} are the same as Bessel functions {{math|''J''''ν''}}, and Weber functions can be expressed as finite linear combinations of [Struve function](/source/Struve_function)s. == Power series expansion == The Anger function has the [power series](/source/power_series) expansion{{dlmf|id=11.10 |title=Anger–Weber Functions |first=R. B. |last=Paris}} : \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)}. While the Weber function has the power series expansion : \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)}. == Differential equations == The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation : z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = 0 . More precisely, the Anger functions satisfy the equation : z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = \frac{(z-\nu)\sin(\pi \nu)}{\pi} , and the Weber functions satisfy the equation : z^2y^{\prime\prime} + zy^\prime +(z^2-\nu^2)y = -\frac{z+\nu+(z-\nu)\cos(\pi \nu)}{\pi}. == Recurrence relations == The Anger function satisfies this inhomogeneous form of [recurrence relation](/source/recurrence_relation) : z\mathbf{J}_{\nu-1}(z)+z\mathbf{J}_{\nu+1}(z)=2\nu\mathbf{J}_\nu(z)-\frac{2\sin\pi\nu}{\pi}. While the Weber function satisfies this inhomogeneous form of [recurrence relation](/source/recurrence_relation) : 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}. == Delay differential equations == The Anger and Weber functions satisfy these homogeneous forms of [delay differential equation](/source/delay_differential_equation)s : \mathbf{J}_{\nu-1}(z)-\mathbf{J}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{J}_\nu(z), : \mathbf{E}_{\nu-1}(z)-\mathbf{E}_{\nu+1}(z)=2\dfrac{\partial}{\partial z}\mathbf{E}_\nu(z). The Anger and Weber functions also satisfy these inhomogeneous forms of [delay differential equation](/source/delay_differential_equation)s : 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}, : 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}. == 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 --- Adapted from the Wikipedia article [Anger function](https://en.wikipedia.org/wiki/Anger_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Anger_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.