Difference between revisions of "Derivative of delta cosine"
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&= \dfrac{ip}{2} (e_{ip}-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)) \\ | &= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\ | ||
| − | &= -\sin_p(t,t_0) | + | &= -\sin_p(t,t_0), |
\end{array}$$ | \end{array}$$ | ||
| − | as was to be shown. █ | + | as was to be shown. █ |
==References== | ==References== | ||
Revision as of 01:49, 6 February 2023
Theorem
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. █
