Anger function

Plot of the Anger function J v(z) with n=2 from -2-2i to 2+2i

Plot of the Anger function J_ν(z) with n = 2 from −2 − 2i to 2 + 2i

In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as

${𝐉}_{\nu }(z)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\cos (\nu \theta -z\sin \theta )d\theta$

with complex parameter $\nu$ and complex variable ⁠$\mathit{\text{𝑧}}$⁠. It is closely related to the Bessel functions.

The Weber function (also known as Lommel–Weber function), introduced by H. F. Weber (1879), is a closely related function defined by

${𝐄}_{\nu }(z)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}\sin (\nu \theta -z\sin \theta )d\theta$

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

Relation between Weber and Anger functions

Plot of the Weber function E v(z) with n=2 from -2-2i to 2+2i

Plot of the Weber function E_ν(z) with n = 2 from −2 − 2i to 2 + 2i

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 ν is not an integer they can be expressed as linear combinations of each other. If ν is an integer then Anger functions J_ν are the same as Bessel functions J_ν, and Weber functions can be expressed as finite linear combinations of Struve functions.

Power series expansion

The Anger function has the power series expansion

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

While the Weber function has the power series expansion

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

Differential equations

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

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

More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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

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

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