q-Cosine

There are several q-analogs of the cosine function.

The two natural definitions of the -cosine defined by Koekoek and Swarttouw (1998) are given by

			(1)
			(2)
			(3)

where and are q-exponential functions. The -cosine and -sine functions satisfy the relations

			(4)
			(5)

Another definition of the -cosine considered by Gosper (2001) is given by

			(6)
			(7)
			(8)
			(9)

where is a Jacobi theta function and is defined via

(10)

This is an even function of unit amplitude, period , and double and triple angle formulas and addition formulas which are analogous to ordinary sine and cosine. For example,

			(11)
			(12)

where is the q-sine, and is q-pi (Gosper 2001). The -cosine also satisfies

(13)