Difference between revisions of "Q-exponential e sub q"
From specialfunctionswiki
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The $q$-exponential $e_q$ is defined by the formula | The $q$-exponential $e_q$ is defined by the formula | ||
| − | $$e_q(z) = $$ | + | $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$ |
=Properties= | =Properties= | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$ | ||
| + | where $e_q$ is the [[Q-exponential e|$q$-exponential $E$]] and $(z;q)_{\infty}$ denotes the [[q-Pochhammer symbol]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
{{:Q-Euler formula for e sub q}} | {{:Q-Euler formula for e sub q}} | ||
Revision as of 17:53, 20 May 2015
The $q$-exponential $e_q$ is defined by the formula $$e_q(z) =\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(q;q)_k}.$$
Contents
Properties
Theorem: The following formula holds: $$e_q(z)=\dfrac{1}{(z;q)_{\infty}},$$ where $e_q$ is the $q$-exponential $E$ and $(z;q)_{\infty}$ denotes the q-Pochhammer symbol.
Proof: █
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.
