Difference between revisions of "Q-sin sub q"
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Revision as of 23:45, 3 May 2015
The function $\sin_q$ is defined by $$\sin_q(z)=\dfrac{e_q(iz)-e_q(-iz)}{2i} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kz^{2k+1}}{(q;q)_{2k+1}}.$$
Properties
Theorem
The following formula holds: $$e_q(iz)=\cos_q(z)+i\sin_q(z),$$ where $e_q$ is the $q$-exponential $e_q$, $\cos_q$ is the $q$-$\cos$ function and $\sin_q$ is the $q$-$\sin$ function.
