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I've been using the R-transform to calculate the free convolution of the eigenvalue spectra of two random matrices and I am trying to understand how it works, and in particular how it relates to noncrossing partitions.

As far as I understand it, the R-transform consists of the following steps:

1. Given a probability distribution function $f(t)$ over some domain $D$ (which I usually take to be $\mathbb R$), find its Cauchy transform $$g(s) = \int_D \frac {f(t)} {t-s} dt$$
2. Calculate the functional inverse $g^{-1}(w)$ and subtract $\frac 1 w$ to obtain the R-transform $$r(w) = g^{-1}(w) - \frac 1 w$$

and that the free convolution of two pdfs $f_1 \boxplus f_2$ consists of:

1. Adding the two R-transformed functions $$r_s (w) = r_1(w) + r_2(w)$$
2. Adding $\frac 1 w$ to the sum, then computing the functional inverse of $$g_s^{-1}(w) = r_s(w)+\frac 1 w$$
3. Computing the inverse Cauchy transform using the Plemelj relation $$f_s(t) = \frac 1 \pi \Im g_s (s)$$