by direct integration

a monotonic Lorentzian relaxation shape. If the standard Lorentzian field distribution, with width parameter

then

This process creates a simple Lorentzian distribution for the magnitude of field, and reveals the corresponding component distribution. The Fourier transform of a simple Lorentzian is well known to be an exponential, so by direct integration:

The convolution has preserved the Lorentzian pathology, the second moments diverge, and the initial slope of the relaxation is nonzero. Similarly, convoluting with the standard Gaussian field distribution results in:

where

where . Again the convolution with the step function has turned the magnitude distribution into the functional form normally associated with the component distribution, and also generated its component distribution. In both cases, the component distribution diverges logarithmically as .

2002-09-24