Revision as of 18:56, 24 May 2016

The function $\sin_q$ is defined by $$\sin_q(z)=\dfrac{e_q(iz)-e_q(-iz)}{2i} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kz^{2k+1}}{(q;q)_{2k+1}},$$ where $e_q$ denotes the $q$-exponential $e$ and $(q;q)_{2k+1}$ denotes the $q$-Pochhammer symbol.

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.

Proof

References

References

[1]