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

$$\mathbf{J}_\nu(z)=\frac{1}{\pi} \int_0^\pi \cos (\nu\theta-z\sin\theta) \,d\theta$$

and is closely related to Bessel functions.

The Weber function, introduced by H. F. Weber (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 functions of the second kind.
Relation between Weber and Anger functions

The Anger and Weber functions are related by

$$\sin(\pi \nu)\mathbf{J}_\nu(z) = \cos(\pi\nu)\mathbf{E}_\nu(z)-\mathbf{E}_{-\nu}(z)$$

$$-\sin(\pi \nu)\mathbf{E}_\nu(z) = \cos(\pi\nu)\mathbf{J}_\nu(z)-\mathbf{J}_{-\nu}(z)$$

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.

Differential equations

The Anger and Weber functions are solutions of inhomogenous 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

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

and the Weber functions satisfy the equation

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

References

Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 12", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, pp. 498, ISBN 978-0486612720, MR0167642.
C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
Paris, R. B. (2010), "Anger-Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F. et al., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR2723248
Prudnikov, A.P. (2001), "Anger function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
Prudnikov, A.P. (2001), "Weber function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104
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

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