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

(1) |

Letting
be a Root of Unity, another set of Weber functions is defined as

(2) | |||

(3) | |||

(4) | |||

(5) | |||

(6) |

(Weber 1902, Atkin and Morain 1993), where is the Dedekind Eta Function. The Weber functions satisfy the identities

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) |

(Weber 1902, Atkin and Morain 1993).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Anger and Weber Functions.'' §12.3 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 498-499, 1972.

Atkin, A. O. L. and Morain, F. ``Elliptic Curves and Primality Proving.'' *Math. Comput.* **61**, 29-68, 1993.

Borwein, J. M. and Borwein, P. B. *Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.*
New York: Wiley, pp. 68-69, 1987.

Weber, H. *Lehrbuch der Algebra, Vols. I-II.* New York: Chelsea, pp. 113-114, 1902.

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1999-05-26