Most of the lectures on the Laplace transform :
- define the Laplace transform Lf as a complex-valued function ;
- give the same example of function that does not admit a Laplace transform : f(t) = exp(t²) ;
- give - with no more explanation - the following proof : the integral from zero to infinity of exp(t² - st) is divergent for every real s.
But of course, an integral parameter from zero to infinity of a function g(s,t) may be divergent for every real s and convergent for some complex s.
So, it seems to me that most of the lectures on the Laplace transform are misleading.
And my question is : Why there is no clear explanation in all these lectures ?