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$M_\text{PMHF}$の計算 (12) |
#223に示した理由により、本稿の議論は全て取り消します。
前稿において、(227.2)右辺第2項を(一部の係数を除き)展開すると、
$$
\require{cancel}
\img[-1.35em]{/images/withinseminar.png}\\
\frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}Q_\mathrm{SM}(t)tf_\text{IF}(t)dt
\tag{228.1}
$$
ここで、WolframAlphaによる級数展開を用いると、
integral_0^(τ) (1 - exp(-λ_2 t)) λ_1 exp(-λ_1 t) dt * (τ^-1)
$$ \frac{1}{\tau}\int_0^\tau F_2(t)f_1(t)dt \approx\frac{1}{2}\lambda_1\lambda_2\tau \tag{228.2} $$
integral_0^(τ) (1 - exp(-λ_2 t)) λ_1 exp(-λ_1 t) t dt * (τ^-1)
$$ \frac{1}{\tau}\int_0^\tau F_2(t)tf_1(t)dt \approx\frac{1}{3}\lambda_1\lambda_2\tau^2 \tag{228.3} $$
$$
(228.1)=\frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}\left[(1-K_\text{SM,MPF})F_\mathrm{SM}(t)tf_\text{IF}(t)+K_\text{SM,MPF}F_\mathrm{SM}(u)tf_\text{IF}(t)\right]dt\\
=\frac{1-K_\text{SM,MPF}}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}F_\text{SM}(t)tf_\text{IF}(t)dt+\frac{K_\text{SM,MPF}}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}F_\text{SM}(u)tf_\text{IF}(t)dt\\
\quad\text{s.t. }u:=t\bmod\tau\tag{228.4}
$$
(228.4)右辺第2項を$t=i\tau+u, i=0,1,...,n-1,T_\text{lifetime}=n\tau$として$t$を$u$で表す変数変換を行うと、
$$
\frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}F_\text{SM}(u)tf_\text{IF}(t)dt
=\frac{1}{T_\text{lifetime}}\sum_{i=0}^{n-1}\int_0^\tau F_\text{SM}(u)(i\tau+u)f_\text{IF}(i\tau+u)du\\
=\frac{\tau}{T_\text{lifetime}}\sum_{i=0}^{n-1}ie^{-\lambda_\text{IF}i\tau}\int_0^\tau F_\text{SM}(u)f_\text{IF}(u)du+\frac{1}{T_\text{lifetime}}\sum_{i=0}^{n-1}e^{-\lambda_\text{IF}i\tau}\int_0^\tau F_\text{SM}(u)uf_\text{IF}(u)du\\
=\frac{1}{\bcancel{T_\text{lifetime}}}\left(\frac{1}{3}\lambda_\text{IF}\lambda_\text{SM}\tau^{\bcancel{3}2}\right)\left(\bcancel{\tau}\frac{\bcancel{T_\text{lifetime}}(T_\text{lifetime}-\tau)}{\bcancel{\tau}^\bcancel{2}}+\frac{\bcancel{T_\text{lifetime}}}{\bcancel{\tau}}\right)\\
=\frac{1}{3}\lambda_\text{IF}\lambda_\text{SM}\tau^2(T_\text{lifetime}-\tau+1)
\tag{228.5}
$$
(228.3)を(228.4)の第1項、(228.5)を第2項に用いて、
$$ (228.1)=\frac{1-K_\text{SM,MPF}}{\bcancel{T_\text{lifetime}}} \left(\frac{1}{3}\lambda_\text{IF}\lambda_\text{SM}T_\text{lifetime}^{\bcancel{3}2}\right) +K_\text{SM,MPF} \left(\frac{1}{3}\lambda_\text{IF}\lambda_\text{SM}\tau^2(T_\text{lifetime}-\tau+1)\right) \tag{228.6} $$
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