The Laplace Transform of z(t) = x(t) * y(t) is Z(s) = X(s) . Y(s).
In the demonstration of this, I fail to see why we are allowed to separate a double integral in two independent integrals.
z(t) = x(t) * y(t) = Integral from -inf to +inf [x(u) . y(t-u) du]
Now let us calculate the two-sided(? - the one that goes from -inf to
+inf) Laplace Transform. It is called «bilatère» in French. What is the correct English adjective?Z(s) = Integ from -inf to +inf{Integ from -inf to +inf [x(u).y(t-u).du] . exp(-s.t).dt} I move the closing bracket to the end = Integ from -inf to +inf{Integ from -inf to +inf [x(u).y(t-u).du . exp(-s.t).dt]} I add two exponentials which have 1 as a product = Integ from -inf to +inf{Integ from -inf to +inf [x(u).exp(-u.t).y(t-u).du.exp(-s.t).exp(u.t) dt]} Now, I simplify a bit and I move «du» to the left = Integ from -inf to +inf{Integ from -inf to +inf [x(u).exp(-u.t).y(t-u).du . exp[-s(t-u)].dt]} = Integ from -inf to +inf{Integ from -inf to +inf [x(u).exp(-u.t).du . y(t-u).exp[-s(t-u)].dt]} On the left side, we see an integral with respect to u variable. On the right side, if we are allowed to make dt = d(t-u), we can substitute v for t-u, and we can separate the integrals.
Why are we allowed to make dt = d(t-u)? This assumes that u is a constant while u sweeps from -inf to +inf!
This message does not deal with an engineering problem but with a maths problem. Could you suggest a more suitable newsgroup?
Writing this message without the usual mathetical notation was difficult. Reading it is difficult. Is there a software tool that permits a better rendering of mathematical expressions with only the ASCII character set?
Thank you very much for your enlightenment.