Difference between revisions of "Q-Cos"
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=Properties= | =Properties= | ||
{{:q-Euler formula for E sub q}} | {{:q-Euler formula for E sub q}} | ||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$D_q \mathrm{Cos}_q(az) = -a \mathrm{Sin}_q(az),$$ | ||
| + | where $D_q$ denotes the [[q-difference operator]], $\mathrm{Cos}$ denotes the [[Q-Cos|$q$-Cosine function]], and $\mathrm{Sin}$ denotes the [[Q-Sin|$q$-Sine function]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
=References= | =References= | ||
[http://homepage.tudelft.nl/11r49/documents/as98.pdf] | [http://homepage.tudelft.nl/11r49/documents/as98.pdf] | ||
Revision as of 21:46, 5 September 2015
The function $\mathrm{Cos}_q$ is defined by $$\mathrm{Cos}_q(z)=\dfrac{E_q(iz)+E_q(-iz)}{2},$$ where $E_q$ denotes the $q$-exponential $E$.
Properties
Theorem
The following formula holds: $$E_q(iz)=\mathrm{Cos}_q(z)+i\mathrm{Sin}_q(z),$$ where $E_q$ is the $q$-exponential $E_q$, $\mathrm{Cos}_q$ is the $q$-$\mathrm{Cos}$ function and $\mathrm{Sin}_q$ is the $q$-$\mathrm{Sin}$ function.
Proof
References
Theorem: The following formula holds: $$D_q \mathrm{Cos}_q(az) = -a \mathrm{Sin}_q(az),$$ where $D_q$ denotes the q-difference operator, $\mathrm{Cos}$ denotes the $q$-Cosine function, and $\mathrm{Sin}$ denotes the $q$-Sine function.
Proof: █
