Difference between revisions of "Derivative of delta cosine"
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| − | + | ==Theorem== | |
| − | + | 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. | ||
| − | + | ||
| − | + | ==Proof== | |
| + | Compute | ||
$$\begin{array}{ll} | $$\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) \\ | \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} (e_{ip}(t,t_0) + e_{-ip}(t,t_0) \\ | ||
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\end{array}$$ | \end{array}$$ | ||
as was to be shown. █ | as was to be shown. █ | ||
| − | + | ||
| − | + | ==References== | |
Revision as of 21:26, 9 June 2016
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. █
