引き続き、前稿の続きの計算をします。本稿では次の(358.1)及び(358.2)を求めます。 $$ \frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}R_\text{SM}(t)f_\text{IF}(\color{red}{u})dt,\ \ s.t.\ \ u:=t\bmod\tau\tag{358.1} $$

$$ \frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}R_\text{SM}(\color{red}{u})f_\text{IF}(\color{red}{u})dt,\ \ s.t.\ \ u:=t\bmod\tau\tag{358.2} $$ (358.1)に$R_\text{SM}(i\tau+u)=e^{-\lambda_\text{SM}(i\tau+u)}$及び、$f_\text{IF}(u)=\lambda_\text{IF} e^{-\lambda_\text{IF}u}$を代入し、 $$ (358.1)=\frac{1}{T_\text{lifetime}}\sum_{i=0}^{n-1} \int_{i\tau}^{(i+1)\tau}e^{-\lambda_\text{SM}(i\tau+u)}\lambda_\text{IF}e^{-\lambda_\text{IF}u}du\\ =\frac{\lambda_\text{IF}}{T_\text{lifetime}}\sum_{i=0}^{n-1}e^{-\lambda_\text{SM}i\tau}\int_0^{\tau}e^{-(\lambda_\text{SM}+\lambda_\text{IF})u}du\tag{358.3} $$ ここで、$\sum_{i=0}^{n-1}e^{-\lambda_\text{SM}i\tau}$を計算すると、等比数列の和及びMaclaurin展開の1次近似より、 $$ \require{cancel} \sum_{i=0}^{n-1}e^{-\lambda_\text{SM}i\tau} =\frac{1-e^{-\lambda_\text{SM}T_\text{lifetime}}}{1-e^{-\lambda_\text{SM}\tau}} \approx\frac{\bcancel{\lambda_\text{SM}}T_\text{lifetime}}{\bcancel{\lambda_\text{SM}}\tau} =\frac{T_\text{lifetime}}{\tau}\tag{358.4} $$ これを用いて、 $$ (358.3)=\frac{\lambda_\text{IF}}{\bcancel{T_\text{lifetime}}}\frac{\bcancel{T_\text{lifetime}}}{\tau}\left[\frac{e^{-(\lambda_\text{SM}+\lambda_\text{IF})u}}{-(\lambda_\text{SM}+\lambda_\text{IF})}\right]^{\tau}_0 =\frac{\lambda_\text{IF}}{\tau(\lambda_\text{SM}+\lambda_\text{IF})}\left(1-e^{-(\lambda_\text{SM}+\lambda_\text{IF})\tau}\right)\tag{358.5} $$ 同様にMaclaurin展開の2次近似を用いると、 $$ e^{-\lambda t}\approx1-\lambda t + \frac{1}{2}\lambda^2 t^2 $$ より、 $$ (358.7)\approx\frac{\lambda_\text{IF}}{\bcancel{\tau(\lambda_\text{SM}+\lambda_\text{IF}})} \left(\bcancel{(\lambda_\text{SM}+\lambda_\text{IF})\tau} -\frac{1}{2}(\lambda_\text{SM}+\lambda_\text{IF})^\bcancel{2}\tau^\bcancel{2}\right)\\ =\lambda_\text{IF}\left(1-\frac{1}{2}(\lambda_\text{SM}+\lambda_\text{IF})\tau\right) =\lambda_\text{IF}-\frac{\lambda_\text{IF}}{2}(\lambda_\text{IF}+\lambda_\text{SM})\tau\tag{358.6} $$ 以上から次のように(358.1)の値が求められました。 $$ \frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}R_\text{SM}(t)f_\text{IF}(u)dt\approx\lambda_\text{IF}\left(1-\frac{1}{2}(\lambda_\text{SM}+\lambda_\text{IF})\tau\right),\ \ s.t.\ \ u:=t\bmod\tau\tag{358.7} $$

次に(358.2)は同様に、$R_\text{SM}(i\tau+u)=e^{-\lambda_\text{SM}(i\tau+u)}$及び、$f_\text{IF}(i\tau+u)=\lambda_\text{IF} e^{-\lambda_\text{IF}i\tau+u}$を代入し、 $$ (358.1)=\frac{1}{T_\text{lifetime}}\sum_{i=0}^{n-1} \int_{i\tau}^{(i+1)\tau}e^{-\lambda_\text{SM}(i\tau+u)}\lambda_\text{IF}e^{-\lambda_\text{IF}(i\tau+u)}du\\ =\frac{\lambda_\text{IF}}{T_\text{lifetime}}\sum_{i=0}^{n-1}e^{-(\lambda_\text{SM}+\lambda_\text{IF})i\tau}\int_0^{\tau}e^{-(\lambda_\text{SM}+\lambda_\text{IF})u}du\tag{358.8} $$ ここで、$\sum_{i=0}^{n-1}e^{-(\lambda_\text{SM}+\lambda_\text{IF})i\tau}$を計算すると、等比数列の和及びMaclaurin展開の1次近似より、 $$ \require{cancel} \sum_{i=0}^{n-1}e^{-(\lambda_\text{SM}+\lambda_\text{IF})i\tau} =\frac{1-e^{-(\lambda_\text{SM}+\lambda_\text{IF})T_\text{lifetime}}}{1-e^{-(\lambda_\text{SM}+\lambda_\text{IF})\tau}} \approx\frac{\bcancel{(\lambda_\text{SM}+\lambda_\text{IF})}T_\text{lifetime}}{\bcancel{(\lambda_\text{SM}+\lambda_\text{IF})}\tau} =\frac{T_\text{lifetime}}{\tau}\tag{358.9} $$ これを用いて、 $$ (358.8)=\frac{\lambda_\text{IF}}{\bcancel{T_\text{lifetime}}}\frac{\bcancel{T_\text{lifetime}}}{\tau}\left[\frac{e^{-(\lambda_\text{SM}+\lambda_\text{IF})u}}{-(\lambda_\text{SM}+\lambda_\text{IF})}\right]^{\tau}_0 =\frac{\lambda_\text{IF}}{\tau(\lambda_\text{SM}+\lambda_\text{IF})}\left(1-e^{-(\lambda_\text{SM}+\lambda_\text{IF})\tau}\right)\tag{358.10} $$ 同様にMaclaurin展開の2次近似を用いると、 $$ e^{-\lambda t}\approx1-\lambda t + \frac{1}{2}\lambda^2 t^2 $$ より、 $$ (358.7)\approx\frac{\lambda_\text{IF}}{\bcancel{\tau(\lambda_\text{SM}+\lambda_\text{IF}})} \left(\bcancel{(\lambda_\text{SM}+\lambda_\text{IF})\tau} -\frac{1}{2}(\lambda_\text{SM}+\lambda_\text{IF})^\bcancel{2}\tau^\bcancel{2}\right)\\ =\lambda_\text{IF}\left(1-\frac{1}{2}(\lambda_\text{SM}+\lambda_\text{IF})\tau\right) =\lambda_\text{IF}-\frac{\lambda_\text{IF}}{2}(\lambda_\text{IF}+\lambda_\text{SM})\tau\tag{358.11} $$ 以上から次のように(358.2)の値が求められました。 $$ \frac{1}{T_\text{lifetime}}\int_0^{T_\text{lifetime}}R_\text{SM}(u)f_\text{IF}(u)dt\approx\lambda_\text{IF}\left(1-\frac{1}{2}(\lambda_\text{SM}+\lambda_\text{IF})\tau\right),\ \ s.t.\ \ u:=t\bmod\tau\tag{358.12} $$

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