q-Sine
There are several q-analogs of the sine function.
The two natural definitions of the q-sine defined by Koekoek and Swarttouw (1998) are given by
where e_q(z) and E_q(z) are q-exponential functions. The q-cosine and q-sine functions satisfy the relations
Another definition of the q-sine considered by Gosper (2001) is given by
where theta_1(z,p) is a Jacobi theta function and p is defined via
| (lnp)(lnq)=pi^2. |
(9)
|
This is an odd function of unit amplitude and period 2pi with double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,
where cos_q^*z is the q-cosine and pi_q is q-pi (Gosper 2001).
See also
q-Cosine, q-Exponential Function, q-Factorial, q-PiExplore with Wolfram|Alpha
More things to try:
References
Gosper, R. W. "Experiments and Discoveries in q-Trigonometry." In Symbolic Computation, Number Theory,Special Functions, Physics and Combinatorics. Proceedings of the Conference Held at the University of Florida, Gainesville, FL, November 11-13, 1999 (Ed. F. G. Garvan and M. E. H. Ismail). Dordrecht, Netherlands: Kluwer, pp. 79-105, 2001.Koekoek, R. and Swarttouw, R. F. The Askey-Scheme of Hypergeometric Orthogonal Polynomials and its q-Analogue. Delft, Netherlands: Technische Universiteit Delft, Faculty of Technical Mathematics and Informatics Report 98-17, pp. 18-19, 1998.Referenced on Wolfram|Alpha
q-SineCite this as:
Weisstein, Eric W. "q-Sine." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-Sine.html