Anger function

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

${\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi \int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta }$

with complex parameter ${\displaystyle \nu }$ and complex variable ⁠ ${\displaystyle {\textit {z$ ⁠.^[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

${\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi \int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta }$

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

Relation between Weber and Anger functions#edit

The Anger and Weber functions are related by

${\displaystyle {\begin{aligned}\sin(\pi \nu )\mathbf {J} _{\nu }(z)&=\cos(\pi \nu )\mathbf {E} _{\nu }(z)-\mathbf {E} _{-\nu }(z),\\-\sin(\pi \nu )\mathbf {E} _{\nu }(z)&=\cos(\pi \nu )\mathbf$

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#edit

The Anger function has the power series expansion^[2]

${\displaystyle \mathbf {J} _{\nu }(z)=\cos {\frac {\pi \nu }{2\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k{4^{k}\Gamma \left(k+{\frac {\nu }{2+1\right)\Gamma \left(k-&#1$

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

${\displaystyle \mathbf {E} _{\nu }(z)=\sin {\frac {\pi \nu }{2\sum _{k=0}^{\infty }{\frac {(-1)^{k}z^{2k{4^{k}\Gamma \left(k+{\frac {\nu }{2+1\right)\Gamma \left(k-&#1$

Differential equations#edit

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

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

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

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

and the Weber functions satisfy the equation^[2]

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

Recurrence relations#edit

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

${\displaystyle z\mathbf {J} _{\nu -1}(z)+z\mathbf {J} _{\nu +1}(z)=2\nu \mathbf {J} _{\nu }(z)-{\frac {2\sin \pi \nu }{\pi .}$

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

${\displaystyle z\mathbf {E} _{\nu -1}(z)+z\mathbf {E} _{\nu +1}(z)=2\nu \mathbf {E} _{\nu }(z)-{\frac {2(1-\cos \pi \nu )}{\pi .}$

Delay differential equations#edit

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

${\displaystyle \mathbf {J} _{\nu -1}(z)-\mathbf {J} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z\mathbf {J} _{\nu }(z),}$

${\displaystyle \mathbf {E} _{\nu -1}(z)-\mathbf {E} _{\nu +1}(z)=2{\dfrac {\partial }{\partial z\mathbf {E} _{\nu }(z).}$

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

${\displaystyle z{\dfrac {\partial }{\partial z\mathbf {J} _{\nu }(z)\pm \nu \mathbf {J} _{\nu }(z)=\pm z\mathbf {J} _{\nu \mp 1}(z)\pm {\frac {\sin \pi \nu }{\pi ,}$

${\displaystyle z{\dfrac {\partial }{\partial z\mathbf {E} _{\nu }(z)\pm \nu \mathbf {E} _{\nu }(z)=\pm z\mathbf {E} _{\nu \mp 1}(z)\pm {\frac {1-\cos \pi \nu }{\pi .}$

References#edit

^ Prudnikov, A.P. (2001) [1994], "Anger function", Encyclopedia of Mathematics, EMS Press
^ a b c d e f g h 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

Categories Special functions

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