Anger function

Plot of the Anger function Template:Math with Template:Math from Template:Math to Template:Math

In mathematics, the Anger function, introduced by Template:Harvs, is a function defined as

${𝐉}_{\nu }(z)=\frac{1}{\pi }{\int }_0^{\pi }\cos (\nu \theta -z\sin \theta )d\theta$

with complex parameter $\nu$ and complex variable Template:Tmath. It is closely related to the Bessel functions.

The Weber function (also known as Lommel–Weber function), introduced by Template:Harvs, is a closely related function defined by

${𝐄}_{\nu }(z)=\frac{1}{\pi }{\int }_0^{\pi }\sin (\nu \theta -z\sin \theta )d\theta$

and is closely related to Bessel functions of the second kind.

Relation between Weber and Anger functions

Plot of the Weber function Template:Math with Template:Math from Template:Math to Template:Math

The Anger and Weber functions are related by

$\begin{array}{cc}\sin (\pi \nu ){𝐉}_{\nu }(z)& =\cos (\pi \nu ){𝐄}_{\nu }(z)-{𝐄}_{-\nu }(z),\\ -\sin (\pi \nu ){𝐄}_{\nu }(z)& =\cos (\pi \nu ){𝐉}_{\nu }(z)-{𝐉}_{-\nu }(z),\end{array}$

so in particular if Template:Math is not an integer they can be expressed as linear combinations of each other. If Template:Math is an integer then Anger functions Template:Math are the same as Bessel functions Template:Math, and Weber functions can be expressed as finite linear combinations of Struve functions.

Power series expansion

The Anger function has the power series expansion

${𝐉}_{\nu }(z)=\cos \frac{\pi \nu }{2}\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{z}^{2k}}{4^k\mathrm{\Gamma }(k+\frac{\nu }{2}+1)\mathrm{\Gamma }(k-\frac{\nu }{2}+1)}+\sin \frac{\pi \nu }{2}\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{z}^{2k+1}}{2^{2k+1}\mathrm{\Gamma }(k+\frac{\nu }{2}+\frac{3}{2})\mathrm{\Gamma }(k-\frac{\nu }{2}+\frac{3}{2})}.$

While the Weber function has the power series expansion

${𝐄}_{\nu }(z)=\sin \frac{\pi \nu }{2}\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{z}^{2k}}{4^k\mathrm{\Gamma }(k+\frac{\nu }{2}+1)\mathrm{\Gamma }(k-\frac{\nu }{2}+1)}-\cos \frac{\pi \nu }{2}\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^k{z}^{2k+1}}{2^{2k+1}\mathrm{\Gamma }(k+\frac{\nu }{2}+\frac{3}{2})\mathrm{\Gamma }(k-\frac{\nu }{2}+\frac{3}{2})}.$

Differential equations

The Anger and Weber functions are solutions of inhomogeneous forms of Bessel's equation

$z^2{y}^{″}+z{y}^{\prime }+(z^2-{\nu }^2)y=0.$

More precisely, the Anger functions satisfy the equation

$z^2{y}^{″}+z{y}^{\prime }+(z^2-{\nu }^2)y=\frac{(z-\nu )\sin (\pi \nu )}{\pi },$

and the Weber functions satisfy the equation

$z^2{y}^{″}+z{y}^{\prime }+(z^2-{\nu }^2)y=-\frac{z+\nu +(z-\nu )\cos (\pi \nu )}{\pi }.$

Recurrence relations

The Anger function satisfies this inhomogeneous form of recurrence relation

$z{𝐉}_{\nu -1}(z)+z{𝐉}_{\nu +1}(z)=2\nu {𝐉}_{\nu }(z)-\frac{2\sin \pi \nu }{\pi }.$

While the Weber function satisfies this inhomogeneous form of recurrence relation

$z{𝐄}_{\nu -1}(z)+z{𝐄}_{\nu +1}(z)=2\nu {𝐄}_{\nu }(z)-\frac{2(1-\cos \pi \nu )}{\pi }.$

Delay differential equations

The Anger and Weber functions satisfy these homogeneous forms of delay differential equations

${𝐉}_{\nu -1}(z)-{𝐉}_{\nu +1}(z)=2\frac{\partial }{\partial z}{𝐉}_{\nu }(z),$

${𝐄}_{\nu -1}(z)-{𝐄}_{\nu +1}(z)=2\frac{\partial }{\partial z}{𝐄}_{\nu }(z).$

The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations

$z\frac{\partial }{\partial z}{𝐉}_{\nu }(z)\pm \nu {𝐉}_{\nu }(z)=\pm z{𝐉}_{\nu \mp 1}(z)\pm \frac{\sin \pi \nu }{\pi },$

$z\frac{\partial }{\partial z}{𝐄}_{\nu }(z)\pm \nu {𝐄}_{\nu }(z)=\pm z{𝐄}_{\nu \mp 1}(z)\pm \frac{1-\cos \pi \nu }{\pi }.$

References

• Template:AS ref
• C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
• Template:Springer
• Template:Citation
• H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76