Sorry that it took a while.
I measure a power law decay of the form
As this model is empirical, the parameters are more or less meaningless.
f(t) = 1/(1+t/t0)^beta.
However, you can interprete this function as a sum of exponential functions:
f(t) = int(1/s*(t0*s)^beta*exp(-t0*s)/GAMMA(beta) * exp(-t*s),s=0..infinity)
is the distribution of decay components. In my case, I sometimes want the distribution of decay times, so I simply replace in the integral s by 1/tau and ds by dtau:
invlaplace(f(t),t,s) = g(s) = 1/s*(t0*s)^beta*exp(-t0*s)/GAMMA(beta)
with the distribution
f(t) = int(1/tau*(t0/tau)^beta*exp(-t0/tau)/GAMMA(beta) *exp(-t/tau),tau=0..infinity)
Using h(tau), I can also define some mean decay time
h(tau) = 1/tau*(t0/tau)^beta*exp(-t0/tau)/GAMMA(beta)
m_tau = int(h(tau)*tau,tau=0..infinity) = t0/(beta-1).