Question

I have a graph between 2 functions f and g.
I know it follows a power law function with exponential cutoff.

f(x) = x**(-alpha)*e**(-lambda*x)

How do I find the value of exponent alpha?

Answer 1

If you have sufficiently close x points (for example one every 0.1), you can try the following:

ln(f(x)) = -alpha ln(x) - lambda x
ln(f(x))' = - alpha / x - lambda

So depending on where you have your points:
If you have a lot of points near 0, you can try:

h(x) = x ln(f(x))' = -alpha - lambda x

So the limit of the function h when x goes to 0 is -alpha
If you have large values of x, the function x -> ln(f(x))’ tends toward lambda when x goes to infinity, so you can guess lambda and use pwdyson’s expression.

If you don’t have close x points, the numerical derivative will be very noisy, so I would try to guess lambda as the limit of -ln(f(x)/x for large x’s…

If you don’t have large values, but a large number of x’s, you can try a minimization of

sum_x_i (ln(y_i) + alpha ln(x_i) + lambda x_i) ^2

on both alpha and lambda (I guess It would be more precise than the initial expression)…
It is a simple least square regression (numpy.linalg.lstsq will do the job).
So you have plenty of methods, the one to chose really depends on you inputs.