Difference between revisions of "Derivative of delta cosine"
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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)) \\ | ||
Revision as of 01:50, 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 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. █
