# Calculate the laplace transform of following function, f(t)=(1)/(1+p(t/tau)^(beta)) where beta<1.

clovnerie0q 2022-10-01 Answered
Calculate the laplace transform of following function,
$f\left(t\right)=\frac{1}{1+p\left(t/\tau {\right)}^{\beta }}$
where $\beta <1$. Any ideas of how to go about the calculation?
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## Answers (1)

Matteo Estes
Answered 2022-10-02 Author has 9 answers
The Laplace transform can be evaluated as an integral of a product of two Meijer G-functions. For $\beta >0$, we obtain
${\int }_{0}^{\mathrm{\infty }}\frac{{e}^{-st}}{1+{t}^{\beta }}dt={\int }_{0}^{\mathrm{\infty }}{G}_{1,1}^{1,1}\left({t}^{\beta }|\genfrac{}{}{0}{}{0}{0}\right){G}_{0,1}^{1,0}\left(st|\genfrac{}{}{0}{}{-}{0}\right)dt=\frac{1}{s}{H}_{2,1}^{1,2}\left({s}^{-\beta }|\genfrac{}{}{0}{}{\left(0,1\right),\left(0,\beta \right)}{\left(0,1\right)}\right).$
For $\beta <0$
${\int }_{0}^{\mathrm{\infty }}\frac{{e}^{-st}}{1+{t}^{\beta }}dt={\int }_{0}^{\mathrm{\infty }}{G}_{1,1}^{1,1}\left({t}^{-\beta }|\genfrac{}{}{0}{}{1}{1}\right){G}_{0,1}^{1,0}\left(st|\genfrac{}{}{0}{}{-}{0}\right)dt=\frac{1}{s}{H}_{2,1}^{1,2}\left({s}^{\beta }|\genfrac{}{}{0}{}{\left(1,1\right),\left(0,-\beta \right)}{\left(1,1\right)}\right),$
which, incidentally, is the same as formally extending the first result to negative $\beta$.
The resulting Fox H-function can be converted to a G-function when $\beta$ is rational.
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