 # Identify the function f(t) that has the following Laplace transform, bar(f)(s)=int_0^(oo)f(t)e^(-st)dt=((1+alpha s)/(1+alpha(s−s_0)))^p where alpha,s_0,p are positive parameters. pobi1k 2022-09-13 Answered
Identify the function f(t) that has the following Laplace transform,
$\stackrel{~}{f}\left(s\right)={\int }_{0}^{\mathrm{\infty }}f\left(t\right){e}^{-st}dt={\left(\frac{1+\alpha s}{1+\alpha \left(s-{s}_{0}\right)}\right)}^{p}$
where $\alpha ,{s}_{0},p$ are positive parameters. Any suggestions about how to approach this problem?
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Using Leucippus's hint, we have
${\mathcal{L}}^{-1}\phantom{\rule{negativethinmathspace}{0ex}}\left[\frac{1}{s}{\left(1+\frac{1}{s}\right)}^{p}\right]={L}_{p}\left(-t\right),\phantom{\rule{0ex}{0ex}}{\mathcal{L}}^{-1}\phantom{\rule{negativethinmathspace}{0ex}}\left[{\left(1+\frac{1}{s}\right)}^{p}\right]=\frac{d}{dt}{L}_{p}\left(-t\right)={L}_{p-1}^{\left(1\right)}\left(-t\right)+\delta \left(t\right),\phantom{\rule{0ex}{0ex}}{\mathcal{L}}^{-1}\phantom{\rule{negativethinmathspace}{0ex}}\left[{\left(\frac{1+\alpha s}{1+\alpha \left(s-{s}_{0}\right)}\right)}^{p}\right]={\mathcal{L}}^{-1}\phantom{\rule{negativethinmathspace}{0ex}}\left[{\left(1+\frac{1}{s/{s}_{0}+1/\left(\alpha {s}_{0}\right)-1}\right)}^{p}\right]=\phantom{\rule{0ex}{0ex}}{s}_{0}{e}^{\left({s}_{0}-1/\alpha \right)t}{L}_{p-1}^{\left(1\right)}\left(-{s}_{0}t\right)+\delta \left(t\right),$
where ${L}_{p-1}^{\left(1\right)}$ is the generalized Laguerre function.