Stable Distribution - Series Representation

Series Representation

The stable distribution can be restated as the real part of a simpler integral:(Peach 1981, § 4.5)

Expressing the second exponential as a Taylor series, we have:

where . Reversing the order of integration and summation, and carrying out the integration yields:

which will be valid for x ≠ μ and will converge for appropriate values of the parameters. (Note that the n = 0 term which yields a delta function in x−μ has therefore been dropped.) Expressing the first exponential as a series will yield another series in positive powers of x−μ which is generally less useful.