Difference between revisions of "Derivative of delta cosine"
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<strong>[[Derivative of delta cosine|Proposition]]:</strong> The following formula holds: | <strong>[[Derivative of delta cosine|Proposition]]:</strong> The following formula holds: | ||
$$\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0),$$ | $$\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0),$$ | ||
| − | where $\cos_p$ denotes the [[Delta cosine|$\Delta\cos_p$]] function and $\sin_p$ denotes the [[Delta sine|$\Delta\sin_p$]] function. | + | where $\cos_p$ denotes the [[Delta cosine|$\Delta$-$\cos_p$]] function and $\sin_p$ denotes the [[Delta sine|$\Delta$-$\sin_p$]] function. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> Compute | <strong>Proof:</strong> Compute | ||
Revision as of 18:02, 21 March 2015
Proposition: The following formula holds: $$\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0),$$ where $\cos_p$ denotes the $\Delta$-$\cos_p$ function and $\sin_p$ denotes the $\Delta$-$\sin_p$ function.
Proof: Compute $$\begin{array}{ll} \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} (e_{ip}(t,t_0) + e_{-ip}(t,t_0) \\ &= \dfrac{ip}{2} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\sin_p(t,t_0) \end{array}$$ as was to be shown. █
