Question 4.5: Verify Eq. (4.18)....
Verify Eq. (4.18).
Step-by-Step
Substitute Eq. (4.2) into Eq. (4.1):
k_{\infty} = \frac{{{\int_0^{\infty}}}[(V_{f}\,/\,V)\nu\Sigma_{f}^{f}(E)+(V_{c}\,/\,V)\nu\Sigma_{f}^{c}(E)+(V_{s t}\,/\,V)\nu\Sigma_{f}^{s t}(E)]\varphi(E)d E}{\int_{0}^{\infty}[(V_{f}/V)\Sigma_{a}^{f}(E)+(V_{c}/V)\Sigma_{a}^{c}(E)+(V_{s t}/V)\Sigma_{a}^{s t}(E)]\varphi(E)d E},
Multiply numerator and denominator by V and note that
\nu\Sigma_{f}^{c}(E)=\nu\Sigma_{f}^{st}(E)=0\,;
k_{\infty} = \frac{V_{f}{\int_{0}^{\infty}}\nu \Sigma_{f}^{f}(E)\varphi(E)d E}{V_{f}{\int_{0}^{\infty}}\Sigma_{a}^{f}(E)\varphi(E)d E+V_{c}\int_0^{\infty}\Sigma_{a}^{c}(E)\varphi(E)d E+V_{s t}\int_0^{\infty}\Sigma_{\alpha}^{st}(E)\varphi(E)d E}
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