q-Cosine
There are several q-analogs of the cosine function.
The two natural definitions of the q-cosine 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-cosine considered by Gosper (2001) is given by
where theta_2(z,p) is a Jacobi theta function and p is defined via
| (lnp)(lnq)=pi^2. |
(10)
|
This is an even function of unit amplitude, period 2pi, and double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,
where sin_qz is the q-sine, and pi_q is q-pi (Gosper 2001). The q-cosine also satisfies
See also
q-Exponential Function, q-Factorial, q-Pi, q-SineExplore with Wolfram|Alpha
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-CosineCite this as:
Weisstein, Eric W. "q-Cosine." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/q-Cosine.html