Weber Functions

Although Bessel functions of the second kind are sometimes called Weber functions, Abramowitz and Stegun (1972) define a separate Weber function as

(1)

These function may also be written as

(2)

where is a regularized hypergeometric function.

This function is implemented in the Wolfram Language as WeberE[nu, z] and is an analog of the Anger function.

Special values for real include

			(3)
			(4)
			(5)
			(6)

where is a Struve function.

Letting be a root of unity, another set of Weber functions is defined as

			(7)
			(8)
			(9)
			(10)
			(11)

(Weber 1902, Atkin and Morain 1993), where is the Dedekind eta function and is the half-period ratio. These functions are related to the Ramanujan g- and G-functions and the elliptic lambda function.

The Weber functions satisfy the identities

			(12)
			(13)
			(14)
			(15)
			(16)
			(17)

(Weber 1902, Atkin and Morain 1993).