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$f(t) = a + b \exp(-c \cdot t ^ d) $, where $a,b,c,d$ are constants, and $d$ is power of $t$.

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It boils down to computing $J = \int_0^\infty \exp(-st - t^d)\, dt$. I'll assume $d > 1$. Write as a series in $-s$: $J = \sum_{n=0}^\infty \int_0^\infty \frac{(-s)^n t^n}{n!} \exp(-t^d)\, dt = \sum_{n=0}^\infty \frac{\Gamma((n+1)/d)}{n!\, d} (-s)^n$. I don't know if there is a "closed form" for this, but for each positive integer $d$ it can be written in terms of hypergeometric functions.

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    @J.M.. I know what has happend. In $a+bexp(-ct^d)$, I missed the multiplication between $b$ and $exp(-ct^d)$, also the multiplication between $c$ and $t^d$. Thus, the wrong answer was gotted. Thank you very much! So, the question doesn't have answer?2011-05-16