Weber Functions

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

$${\mathcal E}_\nu(z)={1\over\pi}\int_0^\pi \sin(\nu\theta-z\sin\theta)\,d\theta.$$ (1)

Letting $\zeta_n=e^{2\pi i/m}$ be a Root of Unity, another set of Weber functions is defined as

$$f(z) = {\eta({\textstyle{1\over 2}}(z+1))\over\zeta_{48}\eta(z)}$$ (2)

$$f_1(z) = {\eta({\textstyle{1\over 2}}z)\over\eta(z)}$$ (3)

$$f_2(z) = \sqrt{2} {\eta(2z)\over\eta(z)}$$ (4)

$$\gamma_2 = {f^{24}(z)-16\over f^8(z)}$$ (5)

$$\gamma_3 = {[f^{24}(z)+8][{f_1}^8(z)-{f_2}^8(z)]\over f^8(z)}$$ (6)

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

$$f(z+1) = {f_1(z)\over\zeta_{48}}$$ (7)

$$f_1(z+1) = {f(z)\over\zeta_{48}}$$ (8)

$$f_2(z+1) = \zeta_{24}f_2(z)$$ (9)

$$f\left({-{1\over z}}\right) = f(z)$$ (10)

$$f_1\left({-{1\over z}}\right) = f_2(z)$$ (11)

$$f_2\left({-{1\over z}}\right) = f_1(z)$$ (12)

(Weber 1902, Atkin and Morain 1993).

See also Anger Function, Bessel Function of the Second Kind, Dedekind Eta Function, j-Function, Jacobi Identities, Jacobi Triple Product, Modified Struve Function, Q-Function, Struve Function

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.