Revision as of 23:24, 26 June 2016

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 $E_q$.

Properties

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: █

@@ Line 5: / Line 5: @@
 =Properties=
 [[q-Euler formula for E sub q]]<br />
+[[q-derivative of q-Sine]]<br />
-<div class="toccolours mw-collapsible mw-collapsed">
-<strong>Theorem:</strong> 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-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">