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Anger function

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(Redirected from Lommel–Weber function)
Plot of the Anger function J v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the Anger function J v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

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

with complex parameter and complex variable .[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

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

Relation between Weber and Anger functions

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The Anger and Weber functions are related by

Plot of the Weber function E v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
Plot of the Weber function E v(z) with n=2 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D

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

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The Anger function has the power series expansion[2]

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

Differential equations

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

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

and the Weber functions satisfy the equation[2]

Recurrence relations

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

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

Delay differential equations

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

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

References

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  1. ^ Prudnikov, A.P. (2001) [1994], "Anger function", Encyclopedia of Mathematics, EMS Press
  2. ^ 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.