![]() It turns out that the real part of is what is crucial: if Re > 0, then the absolute value of e t for t = n, n a large positive real, is very small. ![]() If we had f(t) = e kt as a simple example, that gives you the integral of e t dt, whose antiderivative is (1/) e t, and we have to look at the value of that for t = n, and subtract the value for t = 0. So the integrand e -st f(t) dt might or might not converge, and this in part depends on s, which is a constant as far as the integral is concerned - but we don't know what the constant is. If this limit doesn't exist, if it really matters what value of large n we choose and doesn't 'settle down' as n gets really big, then we say the integral 'does not converge' Not all functions have a finite integral across that range, what we mean when we say 'integral from 0 to infinity' is actually 'the limit as n increases to infinity, of the integral from 0 to n of.'. When you do a Laplace transform of f(t) you have to work out the integral from 0 to infinity of e -st f(t) dt.
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