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Lorentzian distribution of Lorentzians

As an example of a case where a field-magnitude distribution with no peak at finite $\vert{\bf B}\vert$ is generated by this kind of convolution, let

\begin{displaymath}\rho _L(a)=\frac{\textrm{w}}{2\pi }\left( \frac 1{a^2+\textrm{w}^2}\right)

be convoluted with a Lorentzian distribution at each site:

P_{LL}(B_i)= \frac{(norm)\ln \left( \frac{B . . . 
 . . . t( \frac{B^2}{\textrm{w}^2}-1\right)
}\right] .

In spite of these expressions not being well defined when Bi=w or B=w, these distributions are smooth through those values. $P_{LL}(\vert{\bf B}\vert)$ is shown in Fig.5. Note that it is finite, nonzero at B=0. The corresponding relaxation function is expressible in sine-integral and cosine-integral functions.

Figure 5:

Jess H. Brewer