Difference between revisions of "Derivative of Delta sine"
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<strong>[[Derivative of Delta sine|Proposition]]:</strong> The following formula holds: | <strong>[[Derivative of Delta sine|Proposition]]:</strong> The following formula holds: | ||
$$\sin_p^{\Delta}(t,t_0)=p(t)\cos_p(t,t_0),$$ | $$\sin_p^{\Delta}(t,t_0)=p(t)\cos_p(t,t_0),$$ | ||
| − | where $\sin_p$ denotes the [[Delta sine|$\Delta \sin_p$]] function and $\cos_p$ denotes the [[Delta cosine|$\Delta\cos_p$]] function. | + | where $\sin_p$ denotes the [[Delta sine|$\Delta$-$\sin_p$]] function and $\cos_p$ denotes the [[Delta cosine|$\Delta$-$\cos_p$]] function. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> Compute | <strong>Proof:</strong> Compute | ||
Revision as of 18:01, 21 March 2015
Proposition: The following formula holds: $$\sin_p^{\Delta}(t,t_0)=p(t)\cos_p(t,t_0),$$ where $\sin_p$ denotes the $\Delta$-$\sin_p$ function and $\cos_p$ denotes the $\Delta$-$\cos_p$ function.
Proof: Compute $$\begin{array}{ll} \sin^{\Delta}_p(t,t_0) &= \dfrac{1}{2i} \dfrac{\Delta}{\Delta t} \left( e_{ip}(t,t_0) - e_{-ip}(t,t_0) \right) \\ &= \dfrac{ip}{2i} ( e_{ip}(t,t_0) + e_{-ip}(t,t_0) ) \\ &= \dfrac{1}{2} (e_{ip}(t,t_0)+e_{-ip}(t,t_0)) \\ &= p\cos_p(t,t_0), \end{array}$$ as was to be shown. █
