Anger function

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 (also known as Lommel-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

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

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 inhomogeneous forms of Bessel's equation

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

More precisely, the Anger functions satisfy the equation

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

and the Weber functions satisfy the equation

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

References

• 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
• 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.
• Prudnikov, A.P. (2001) [1994], "Anger function", Encyclopedia of Mathematics, EMS Press
• Prudnikov, A.P. (2001) [1994], "Weber function", Encyclopedia of Mathematics, EMS Press
• 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