the R- and S-transforms
The R- and S-transforms are to free probability what the logarithm of the characteristic function is to classical probability: they are the transforms that linearise convolution. Computing the free additive convolution of two spectra directly from the definition is hopeless; the genius of Voiculescu's theory is that there is a recipe — pass to a transform, add (or multiply), invert — that makes free convolution as mechanical as adding cumulants. The R-transform linearises free addition; the S-transform linearises free multiplication.
Start from the Stieltjes (Cauchy) transform G(z) = integral of 1/(z - x) dmu(x) of a measure mu (note the sign convention; this is the standard free-probability convention). Its functional inverse is the K-function: K(G(z)) = z. The R-transform is defined by R(w) = K(w) - 1/w, that is, R(w) = G^{-1}(w) - 1/w. The defining miracle is additivity: if mu and nu are free, then R_{mu box-plus nu}(w) = R_mu(w) + R_nu(w). So to add two free spectra you compute each R-transform, add them as functions, then invert the chain back through K and G to recover the convolved measure. The R-transform's power-series coefficients are the free cumulants, the free analogues of classical cumulants; freeness is the statement that mixed free cumulants vanish, exactly as independence is the vanishing of mixed classical cumulants. For multiplication there is the S-transform: starting from the moment generating series psi(z) = sum of tau(a^k) z^k, one forms chi as its inverse and sets S(z) = (1 + z)/z times chi(z); its defining property is multiplicativity, S_{mu box-times nu} = S_mu times S_nu for free mu, nu (with mild positivity assumptions).
These transforms make free probability computational. To find the spectrum of a sum of free matrices: R-transforms add. To find the spectrum of a product (or of A^{1/2} B A^{1/2}): S-transforms multiply. The semicircle has the simplest possible R-transform, R(w) = w (which is why a sum of free semicirculars is semicircular with added variances — the free CLT in one line), and the Marchenko-Pastur and free-Poisson laws have clean S-transforms, which is how the deformed MP law for structured covariance is derived. A caveat: the additivity and multiplicativity of these transforms hold only under freeness, and only as formal/analytic identities valid where the transforms are defined (the Cauchy transform lives off the support, and inverting it requires care near the spectral edges). The R-transform linearises free, not classical, convolution; applying it to genuinely independent commuting variables gives the wrong distribution. And the S-transform's standard form needs the measure to have nonzero mean (a positive support), since one divides by the first moment.
The semicircle law has R-transform R(w) = w — the simplest non-trivial example. Adding n free copies of a (rescaled) semicircle therefore gives R-transform n times w/sqrt(n)^... ; concretely, normalising a sum of n free identically-distributed centred elements by 1/sqrt(n) makes the R-transforms add to (in the limit) the linear function characterising a semicircle. That single line — R-transforms add, the semicircle's is linear — is the entire proof of the free central limit theorem.
R-transforms add under free convolution; the semicircle's is linear, giving the free CLT in one step.
The R-transform linearises FREE convolution, not classical; do not apply it to ordinary independent variables. The S-transform's standard definition divides by the first moment, so it needs a measure with nonzero mean (positive support) to be well posed.