Anger function explained

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

J _{\nu(z) = (1)/(\pi)} \int ^{\pi} _{0} \cos (\nu\theta - z\sin\theta) d\theta

with complex parameter

\nu and complex variable it{z} . It is closely related to the Bessel functions.

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

E _{\nu(z) = (1)/(\pi)} \int ^{\pi} _{0} \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

The Anger and Weber functions are related by

\begin{align} \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{align}

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 (\pi\nu)/(2)} \sum ^{inf ty (( -1 ) ^{kz} ^{2k})/(4 ^{k\Gamma\left(k + (\nu)/(2)} +1 \right)\Gamma\left(k - (\nu)/(2) +1 \right))} _{k =0} + \sin (\pi\nu)/(2) \sum ^{inf ty (( -1 ) ^{kz} ^{2k+1})/(2 ^{2 k +1} \Gamma\left(k + (\nu)/(2) + (3)/(2) \right)\Gamma\left(k - (\nu)/(2) + (3)/(2) \right))} _{k =0} .

While the Weber function has the power series expansion

E _{\nu(z) = \sin (\pi\nu)/(2)} \sum ^{inf ty (( -1 ) ^{kz} ^{2k})/(4 ^{k\Gamma\left(k + (\nu)/(2)} +1 \right)\Gamma\left(k - (\nu)/(2) +1 \right))} _{k =0} - \cos (\pi\nu)/(2) \sum ^{inf ty (( -1 ) ^{kz} ^{2k+1})/(2 ^{2 k +1} \Gamma\left(k + (\nu)/(2) + (3)/(2) \right)\Gamma\left(k - (\nu)/(2) + (3)/(2) \right))} _{k =0} .

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} = ((z - \nu)\sin(\pi \nu))/(\pi) ,

and the Weber functions satisfy the equation

z ^{2y} ^{\prime\prime} + zy ^{\prime} + (z ^{2-\nu} ^{2)y} =- (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) - (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) - (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 \dfrac{\partial}{\partial z} J _{\nu(z),} E _{\nu-1} (z) - E _{\nu+1} (z) =2 \dfrac{\partial}{\partial z} E _{\nu(z).}

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

z\dfrac{\partial}{\partial z} J _{\nu(z)\pm\nuJ} _{\nu(z)=\pm} z J _{\nu\mp1} (z)\pm (\sin\pi\nu)/(\pi) , z\dfrac{\partial}{\partial z} E _{\nu(z)\pm\nuE} _{\nu(z)=\pm} z E _{\nu\mp1} (z)\pm (1- \cos\pi\nu)/(\pi) .

References

• • C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
  • G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952)
• H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76