Chapter07 - Home

Exercise. This is the solution to exercise 7.1 in the book.

Solution. We can write $V$ as

$$V={G}_{0}{\left(E\right)}^{-1}-G{\left(E\right)}^{-1}$$If we replace $V$ in ${G}_{0}\left(E\right)VG\left(E\right)$ we get:

$${G}_{0}\left(E\right)VG\left(E\right)={G}_{0}\left(E\right)\left({G}_{0}{\left(E\right)}^{-1}-G{\left(E\right)}^{-1}\right)G\left(E\right)$$or

$${G}_{0}\left(E\right)VG\left(E\right)={G}_{0}\left(E\right){G}_{0}{\left(E\right)}^{-1}G\left(E\right)-{G}_{0}\left(E\right)G{\left(E\right)}^{-1}G\left(E\right)$$or

$${G}_{0}\left(E\right)VG\left(E\right)=G\left(E\right)-{G}_{0}\left(E\right)$$Therefore

$$G\left(E\right)={G}_{0}\left(E\right)+{G}_{0}\left(E\right)VG\left(E\right)$$