I have been reading a book on automatic control and got stucked in the derivation of the laplace transform of the convolution integral. The autor says:
Laplace transform of the convolution of $g$ and $u$: \begin{align*} &\hat{y}(s)=\int_{t=0}^\infty \left[ \int_{\tau =0}^\infty g(t-\tau) u(\tau)d\tau \right] e^{-st}dt \end{align*} Interchanching order of integrations: \begin{align*} &\hat{y}(s)=\int_{t=0}^\infty \left[ \int_{\tau =0}^\infty g(t-\tau) u(\tau)d\tau \right] e^{-s(t-\tau)}e^{-s\tau}dt=\int_{\tau=0}^\infty \left[ \int_{t=0}^\infty g(t-\tau)e^{-s(t-\tau)}dt \right] u(\tau) e^{-s\tau}d\tau \end{align*}
My question is: Is it correct to put the $e^{-s(t-\tau)}$ term in the inner integral in the way he dit it?.
I found another way of deriving this result wich I think is correct: https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/transfer-system-and-weight-functions-greens-formula/MIT18_03SCF11_s30_5text.pdf
Thanks in advance, hope I'm not misunderstanding the procedure in the book.