Difference between revisions of "Q-Sin"
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$$D_q \mathrm{Sin}_q(bz) = b \mathrm{Cos}_q(bz),$$ | $$D_q \mathrm{Sin}_q(bz) = b \mathrm{Cos}_q(bz),$$ | ||
where $D_q$ is the [[q-difference operator]], $\mathrm{Sin}_q$ is the [[Q-Sin|$q$-Sine function]], and $\mathrm{Cos}_q$ is the [[Q-Cos|$q$-cosine function]]. | where $D_q$ is the [[q-difference operator]], $\mathrm{Sin}_q$ is the [[Q-Sin|$q$-Sine function]], and $\mathrm{Cos}_q$ is the [[Q-Cos|$q$-cosine function]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The [[general solution]] of the [[q-difference equation|$q$-difference equation]] $D_q^2 y(x) + k^2 y(x) = 0$ | ||
| + | is $y(x)=c_1 \mathrm{Cos}_q(kz) + c_2 \mathrm{Sin}_q(kz).$ | ||
| + | |||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 21:49, 5 September 2015
The function $\mathrm{Sin}_q$ is defined by $$\mathrm{Sin}_q(z)=\dfrac{E_q(iz)-E_q(-iz)}{2i},$$ 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: $$D_q \mathrm{Sin}_q(bz) = b \mathrm{Cos}_q(bz),$$ where $D_q$ is the q-difference operator, $\mathrm{Sin}_q$ is the $q$-Sine function, and $\mathrm{Cos}_q$ is the $q$-cosine function.
Proof: █
Theorem: The general solution of the $q$-difference equation $D_q^2 y(x) + k^2 y(x) = 0$ is $y(x)=c_1 \mathrm{Cos}_q(kz) + c_2 \mathrm{Sin}_q(kz).$
Proof: █
