Sorry that it took a while.

One sample:

I measure a power law decay of the form

Code:

f(t) = 1/(1+t/t0)^beta.

As this model is empirical, the parameters are more or less meaningless.

However, you can interprete this function as a sum of exponential functions:

Code:

f(t) = int(1/s*(t0*s)^beta*exp(-t0*s)/GAMMA(beta) * exp(-t*s),s=0..infinity)

Where

Code:

invlaplace(f(t),t,s) = g(s) = 1/s*(t0*s)^beta*exp(-t0*s)/GAMMA(beta)

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:

Code:

f(t) = int(1/tau*(t0/tau)^beta*exp(-t0/tau)/GAMMA(beta) *exp(-t/tau),tau=0..infinity)

with the distribution

Code:

h(tau) = 1/tau*(t0/tau)^beta*exp(-t0/tau)/GAMMA(beta)

Using h(tau), I can also define some mean decay time

Code:

m_tau = int(h(tau)*tau,tau=0..infinity) = t0/(beta-1).

Alex