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Erschienen in:
2017 | OriginalPaper | Buchkapitel
3. Free Harmonic Analysis
verfasst von : James A. Mingo, Roland Speicher
Erschienen in: Free Probability and Random Matrices
Verlag: Springer New York
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Abstract
In this chapter we shall present an approach to free probability based on analytic functions. At the end of the previous chapter, we defined the Cauchy transform of a random variable a in an algebra \(\mathcal{A}\) with a state φ to be the formal power series \(G(z) = \frac{1} {z}M(\frac{1} {z})\) where M(z) = 1 + ∑
n ≥ 1
α
n
z
n
and α
n
= φ(a
n
) are the moments of a. Then R(z), the R-transform of a, was defined to be the formal power series R(z) = ∑
n ≥ 1
κ
n
z
n−1 determined by the moment-cumulant relation which we have shown to be equivalent to the equations
$$\displaystyle{G\big(R(z) + 1/z\big) = z = 1/G(z) + R(G(z)).}$$