## On the existence of the Laplace Transform [closed]

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 ?

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I fear this is both not a real question AND subjective and argumentative, so I am voting to close. In addition, can you find a complex $s$ for which the integral converges? – Igor Rivin Jan 12 2012 at 15:22
I second Igor's comment. Furthermore, you may consider to post a reformulation your question at math.stackexchange.com. – Dirk Jan 12 2012 at 15:27