Difference between revisions of "Derivative of delta cosine"
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The following formula holds: | 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 $\ | + | where $\cos^{\Delta}_p$ denotes the [[Delta derivative|delta derivative]] of the [[Delta cosine|delta cosine]] function and $\sin_p$ denotes the [[Delta sine|delta sine]] function. |
==Proof== | ==Proof== | ||
Compute | 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} \Big(e_{ip}(t,t_0) + e_{-ip}(t,t_0) \Big) \\ |
&= \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== | ||
Latest revision as of 01:51, 6 February 2023
Theorem
The following formula holds: $$\cos_p^{\Delta}(t,t_0)=-p(t)\sin_p(t,t_0),$$ where $\cos^{\Delta}_p$ denotes the delta derivative of the delta cosine function and $\sin_p$ denotes the delta sine function.
Proof
Compute $$\begin{array}{ll} \cos_p^{\Delta}(t,t_0) &= \dfrac{1}{2} \dfrac{\Delta}{\Delta t} \Big(e_{ip}(t,t_0) + e_{-ip}(t,t_0) \Big) \\ &= \dfrac{ip}{2} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\dfrac{p}{2i} (e_{ip}-e_{-ip}(t,t_0)) \\ &= -\sin_p(t,t_0), \end{array}$$ as was to be shown. █
