# Derivation of delta e sub p on T=Z

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If $t>s$, then $$\begin{array}{ll} e_p(t,s) &= \exp \left( \displaystyle\int_{s}^{t} \dfrac{1}{\mu(\tau)} \log(1 + p(\tau)) \Delta \tau \right) \\ &= \exp \left( \displaystyle\sum_{k=s}^{t-1} \log(1+p(k)) \right) \\ &= \displaystyle\prod_{k=s}^{t-1} \left( 1+p(k) \right). \\ \end{array}$$ If $t<s$ then the integral $\displaystyle\int_s^t \dfrac{1}{\mu(\tau)} \log(1+p(\tau))\Delta \tau = -\displaystyle\int_t^s \dfrac{1}{\mu(\tau)}\log(1+p(\tau))\Delta \tau$, and so we see $$e_p(t,s) = \dfrac{1}{e_p(s,t)}.$$