$f(t) = a + b \exp(-c \cdot t ^ d) $, where $a,b,c,d$ are constants, and $d$ is power of $t$.
Can anyone help me get this Laplace transform of
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integral-transforms
1 Answers
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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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1Apparently, as long as $d$ is rational and bigger than 1, the expression is hypergeometric. The expressions are rather unwieldy here that a numerical method might make more sense for computational needs. – 2011-05-16
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0Thanks for the answer given by @Robert Israel and @J.M.. But the answer of the question is $(a+bexp(−c*t^d))/s$ computed by the mathematic tool of Maple. As you find, the answer contains $t$. So, what is the right answer of the problem? – 2011-05-16
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0@zghu: What, \*exactly\*, did you input into Maple? – 2011-05-16
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0@J.M.. I have put $a+bexp(-ct^d)$ into Maple, and got that result. – 2011-05-16
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0@zghu: I mean, could you paste in here the exact input you gave to Maple? I have a feeling your syntax is wrong somewhere. – 2011-05-16
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1@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