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I wish to calculate the laplace transform of following function,

$$ f(t) = \frac{1}{1+p (t/\tau)^\beta} $$ where $\beta < 1$. Any ideas of how to go about the calculation?

user35952
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1 Answers1

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The Laplace transform can be evaluated as an integral of a product of two Meijer G-functions. For $\beta > 0$, we obtain $$\int_0^\infty \frac {e^{-s t}} {1 + t^\beta} dt = \int_0^\infty G_{1, 1}^{1, 1} \left( t^\beta \middle| {0 \atop 0} \right) G_{0, 1}^{1, 0} \left( s t \middle| {- \atop 0} \right) dt = \frac 1 s H_{2, 1}^{1, 2} \left( s^{-\beta} \middle| {(0, 1), (0, \beta) \atop (0, 1) } \right).$$ For $\beta < 0$, $$\int_0^\infty \frac {e^{-s t}} {1 + t^\beta} dt = \int_0^\infty G_{1, 1}^{1, 1} \left( t^{-\beta} \middle| {1 \atop 1} \right) G_{0, 1}^{1, 0} \left( s t \middle| {- \atop 0} \right) dt = \frac 1 s H_{2, 1}^{1, 2} \left( s^\beta \middle| {(1, 1), (0, -\beta) \atop (1, 1) } \right),$$ 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.

Maxim
  • 10,764
  • This is an interesting insight! Thank you! – user35952 Sep 26 '18 at 06:33
  • I have a doubt about this result. I think it is valid only if $\beta=1$. Could you derive step by step? – user48672 Nov 20 '22 at 12:44
  • @user48672 If you're testing the result numerically, what values of $\beta$ and $s$ are you taking? Most likely the issue is in the numerical evaluation of the H-function. To evaluate the integral, we find $\mathcal M[t \mapsto e^{-t}]$, which gives Euler's integral of the second kind, and $\mathcal M[t \mapsto 1/(1 + t^\beta)]$, which reduces to Euler's integral of the first kind, and apply the Mellin convolution method. – Maxim Nov 20 '22 at 21:13