Difference between revisions of "Q-sin sub q"
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(Created page with "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= {{:q-Eule...") |
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The function $\sin_q$ is defined by | The function $\sin_q$ is defined by | ||
| − | $$\sin_q(z)=\dfrac{e_q(iz)-e_q(-iz)}{2i} = \displaystyle\sum_{k=0}^{\infty} \dfrac{(-1)^kz^{2k+1}}{(q;q)_{2k+1}} | + | $$\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}},$$ |
| + | where $e_q$ denotes the [[q-exponential e|$q$-exponential $e$]] and $(q;q)_{2k+1}$ denotes the [[q-Pochhammer|$q$-Pochhammer symbol]]. | ||
=Properties= | =Properties= | ||
Revision as of 23:50, 3 May 2015
The function $\sin_q$ is defined by $$\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}},$$ where $e_q$ denotes the $q$-exponential $e$ and $(q;q)_{2k+1}$ denotes the $q$-Pochhammer symbol.
Properties
Theorem
The following formula holds: $$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.
