Stretched Exponential Function

The stretched exponential function, also called the Kohlrausch or Kohlrausch-Williams-Watts function, is the function

(1)

where , , and conventionally (Williams and Watts 1970). The case is the ordinary exponential function.

For , the function is completely monotonic and hence is a Laplace transform. More explicitly,

(2)

where is the one-sided stable distribution with stability index (Pollard 1946). Consequently, its exact probability density over exponential relaxation times is

			(3)
			(4)

For example, when ,

(5)

The improper integral of the stretched exponential, also interpreted as its mean relaxation time, is

(6)

This should not be confused with using a normalized stretched exponential itself as a phenomenological probability density,

(7)

which is generally different from the exact relaxation-time probability density (Ribeiro de Almeida et al. 2026).