Anger function

From Wikipedia, the free encyclopedia

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

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

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

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 Jν are the same as Bessel functions Jν, and Weber functions can be expressed as finite linear combinations of Struve functions.

Power series expansion[]

The Anger function has the power series expansion[1]

While the Weber function has the power series expansion[1]

Differential equations[]

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

More precisely, the Anger functions satisfy the equation[1]

and the Weber functions satisfy the equation[1]

Recurrence relations[]

The Anger function satisfies this inhomogeneous form of recurrence relation[1]

While the Weber function satisfies this inhomogeneous form of recurrence relation[1]

Delay differential equations[]

The Anger and Weber functions satisfy these homogeneous forms of delay differential equations[1]

The Anger and Weber functions also satisfy these inhomogeneous forms of delay differential equations[1]

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
  • 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
  1. ^ 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
Retrieved from ""