Anger function

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

[1] It is closely related to the Bessel 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.

The Anger function has the power series expansion[2] While the Weber function has the power series expansion[2] 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] The Anger function satisfies this inhomogeneous form of recurrence relation[2] While the Weber function satisfies this inhomogeneous form of recurrence relation[2] 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]

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
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