Difference between revisions of "Derivation of delta sin sub p for T=Z"
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(Created page with "$$\begin{array}{ll} \sin_p(t,s) &= \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} \\ &= \dfrac{\displaystyle\prod_{k=t_0}^{t-1}1+ip(k) - \displaystyle\prod_{k=t_0}^{t-1}1-ip(k)}{2i} \en...") |
m (Tom moved page Derivation of delta sin sub p on T=Z to Derivation of delta sin sub p for T=Z) |
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| − | $$\ | + | Using the properties of [[Delta exponential|$e_p$]], it is clear that $\sin_p(t,s) = \dfrac{1-1}{2i} = 0$. Furthermore if $t>s$, then |
| − | \sin_p(t,s) | + | $$\sin_p(t,s) = \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} = \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i}.$$ |
| − | + | If $t<s$ then | |
| − | \ | + | $$\sin_p(t,s) = \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} = \dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i}.$$ |
Latest revision as of 00:46, 22 May 2015
Using the properties of $e_p$, it is clear that $\sin_p(t,s) = \dfrac{1-1}{2i} = 0$. Furthermore if $t>s$, then $$\sin_p(t,s) = \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} = \dfrac{\displaystyle\prod_{k=s}^{t-1}1+ip(k) - \displaystyle\prod_{k=s}^{t-1}1-ip(k)}{2i}.$$ If $t<s$ then $$\sin_p(t,s) = \dfrac{e_{ip}(t,s)-e_{-ip}(t,s)}{2i} = \dfrac{\displaystyle\prod_{k=t}^{s-1} \frac{1}{1+ip(k)} - \displaystyle\prod_{k=t}^{s-1} \frac{1}{1-ip(k)}}{2i}.$$
