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

\mathbf_\nu(z)=\frac \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta

and is closely related to

Bessel function Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

s. The Weber function (also known as

Lommel Lommel () is a municipality and a city in the Belgian province of Limburg. The Kempen city has about 34,000 inhabitants and is part of the electoral district and the judicial district Lommel Neerpelt. Besides residential town of Lommel also has ...

–Weber function), introduced by , is a closely related function defined by :

\mathbf_\nu(z)=\frac \int_0^\pi \sin (\nu\theta-z\sin\theta) \,d\theta

and is closely related to

s of the second kind.

Relation between Weber and Anger functions

The Anger and Weber functions are related by :

\begin
\sin(\pi \nu)\mathbf_\nu(z) &= \cos(\pi\nu)\mathbf_\nu(z)-\mathbf_(z), \\
-\sin(\pi \nu)\mathbf_\nu(z) &= \cos(\pi\nu)\mathbf_\nu(z)-\mathbf_(z),
\end

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 function In mathematics, the Struve functions , are solutions of the non-homogeneous Bessel's differential equation: : x^2 \frac + x \frac + \left (x^2 - \alpha^2 \right )y = \frac introduced by . The complex number α is the order of the Struve functio ...

Power series expansion

The Anger function has the power series expansion :

\mathbf_\nu(z)=\cos\frac\sum_^\infty\frac+\sin\frac\sum_^\infty\frac.

While the Weber function has the power series expansion :

\mathbf_\nu(z)=\sin\frac\sum_^\infty\frac-\cos\frac\sum_^\infty\frac.

Differential equations

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

z^2y^ + zy^\prime +(z^2-\nu^2)y = 0 .

More precisely, the Anger functions satisfy the equation :

z^2y^ + zy^\prime +(z^2-\nu^2)y = \frac ,

and the Weber functions satisfy the equation :

z^2y^ + zy^\prime +(z^2-\nu^2)y = -\frac.

Recurrence relations

The Anger function satisfies this inhomogeneous form of

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

z\mathbf_(z)+z\mathbf_(z)=2\nu\mathbf_\nu(z)-\frac.

While the Weber function satisfies this inhomogeneous form of

z\mathbf_(z)+z\mathbf_(z)=2\nu\mathbf_\nu(z)-\frac.

Delay differential equations

The Anger and Weber functions satisfy these homogeneous forms of

delay differential equation In mathematics, delay differential equations (DDEs) are a type of differential equation in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. DDEs are also called time ...

s :

\mathbf_(z)-\mathbf_(z)=2\dfrac\mathbf_\nu(z),

\mathbf_(z)-\mathbf_(z)=2\dfrac\mathbf_\nu(z).

The Anger and Weber functions also satisfy these inhomogeneous forms of

s :

z\dfrac\mathbf_\nu(z)\pm\nu\mathbf_\nu(z)=\pm z\mathbf_(z)\pm\frac,

z\dfrac\mathbf_\nu(z)\pm\nu\mathbf_\nu(z)=\pm z\mathbf_(z)\pm\frac.

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 {{Reflist Special functions