Anger function

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

𝐉_{ν} (z) = \frac{1}{π} \int_{0}^{π} \cos (ν θ - z \sin θ) d θ

with complex parameter $ν$ and complex variable ⁠ $𝑧$ ⁠.^[1] 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_{ν} (z) = \frac{1}{π} \int_{0}^{π} \sin (ν θ - z \sin θ) d θ

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{matrix} \sin (π ν) 𝐉_{ν} (z) & = \cos (π ν) 𝐄_{ν} (z) - 𝐄_{- ν} (z), \\ - \sin (π ν) 𝐄_{ν} (z) & = \cos (π ν) 𝐉_{ν} (z) - 𝐉_{- ν} (z), \end{matrix}

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^[2]

𝐉_{ν} (z) = \cos \frac{π ν}{2} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k}}{4^{k} Γ (k + \frac{ν}{2} + 1) Γ (k - \frac{ν}{2} + 1)} + \sin \frac{π ν}{2} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k + 1}}{2^{2 k + 1} Γ (k + \frac{ν}{2} + \frac{3}{2}) Γ (k - \frac{ν}{2} + \frac{3}{2})} .

While the Weber function has the power series expansion^[2]

𝐄_{ν} (z) = \sin \frac{π ν}{2} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k}}{4^{k} Γ (k + \frac{ν}{2} + 1) Γ (k - \frac{ν}{2} + 1)} - \cos \frac{π ν}{2} \sum_{k = 0}^{\infty} \frac{(- 1)^{k} z^{2 k + 1}}{2^{2 k + 1} Γ (k + \frac{ν}{2} + \frac{3}{2}) Γ (k - \frac{ν}{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^{'} + (z^{2} - ν^{2}) y = 0.

More precisely, the Anger functions satisfy the equation^[2]

z^{2} y^{″} + z y^{'} + (z^{2} - ν^{2}) y = \frac{(z - ν) \sin (π ν)}{π},

and the Weber functions satisfy the equation^[2]

z^{2} y^{″} + z y^{'} + (z^{2} - ν^{2}) y = - \frac{z + ν + (z - ν) \cos (π ν)}{π} .

Recurrence relations

The Anger function satisfies this inhomogeneous form of recurrence relation^[2]

z 𝐉_{ν - 1} (z) + z 𝐉_{ν + 1} (z) = 2 ν 𝐉_{ν} (z) - \frac{2 \sin π ν}{π} .

While the Weber function satisfies this inhomogeneous form of recurrence relation^[2]

z 𝐄_{ν - 1} (z) + z 𝐄_{ν + 1} (z) = 2 ν 𝐄_{ν} (z) - \frac{2 (1 - \cos π ν)}{π} .

Delay differential equations

The Anger and Weber functions satisfy these homogeneous forms of delay differential equations^[2]

𝐉_{ν - 1} (z) - 𝐉_{ν + 1} (z) = 2 \frac{\partial}{\partial z} 𝐉_{ν} (z),

𝐄_{ν - 1} (z) - 𝐄_{ν + 1} (z) = 2 \frac{\partial}{\partial z} 𝐄_{ν} (z) .

The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations^[2]

z \frac{\partial}{\partial z} 𝐉_{ν} (z) \pm ν 𝐉_{ν} (z) = \pm z 𝐉_{ν \mp 1} (z) \pm \frac{\sin π ν}{π},

z \frac{\partial}{\partial z} 𝐄_{ν} (z) \pm ν 𝐄_{ν} (z) = \pm z 𝐄_{ν \mp 1} (z) \pm \frac{1 - \cos π ν}{π} .

References

↑ Prudnikov, A.P. (2001) [1994], "Anger function", Encyclopedia of Mathematics, EMS Press
1 2 3 4 5 6 7 8 Paris, R. B. (2010), "Anger–Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 12". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 498. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.
C.T. Anger, Neueste Schr. d. Naturf. d. Ges. i. Danzig, 5 (1855) pp. 1–29
Prudnikov, A.P. (2001) [1994], "Weber function", Encyclopedia of Mathematics, EMS Press
Watson, G.N. (1952), A treatise on the theory of Bessel functions, Cambridge Univ. Press, pp. 1–2
H.F. Weber, Zurich Vierteljahresschrift, 24 (1879) pp. 33–76

[EOM_Anger-1] Prudnikov, A.P. (2001) [1994], "Anger function", Encyclopedia of Mathematics, EMS Press

[DLMF-2] 1 2 3 4 5 6 7 8 Paris, R. B. (2010), "Anger–Weber Functions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.

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