Narrow Results

Muraki has shown that monotonic convolution of probability measures corresponds with composition of the reciprocal Cauchy transforms corresponding with these measures. We provide a new proof of this fact, not involving the combinatorics of moments.

Let g : ℝ → ℂ be a C^∞-function with all derivatives bounded and let tr_n denote the normalized trace on the n × n matrices. In Ref. 3 Ercolani and McLaughlin established asymptotic expansions of the mean value 𝔼{tr_n(g(X_n))} for a rather general class of random matrices X_n, including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a random matrix X_n that

Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection. Let V(z, N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N. Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z, N) that satisfies a jump condition on the contour Γ with the jump matrix V(z, N). Assume that V(z, N) has an asymptotic expansion, as N → ∞, on Γ. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z, N), as n → ∞, for z ∈ ℂ\Γ. Our method makes use of only complex analysis.

Formulas are derived for expressing Cauchy and Hilbert transforms of a function f in terms of Cauchy and Hilbert transforms of f(x^r). When r is an integer, this corresponds to evaluating the Cauchy transform of f(x^r) at all choices of z^1/r. Related formulas for rational r result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard Cauchy transform. These formulas are used to regularize the behavior of functions that are slowly decaying or oscillatory, in order to facilitate numerical computation and extend asymptotic results.

A differential and integral criterion for monogenicity is presented within the framework of Clifford analysis. The results obtained provide characterizations for the holomorphic and the two-sided biregular functions.

Let Γ be a piecewise smooth contour in ℂ, which could be unbounded and may have points of self-intersection. Let V (z,N) be a 2 × 2 matrix-valued function defined on Γ, which depends on a parameter N. Consider a Riemann–Hilbert problem for a matrix-valued analytic function R(z,N) that satisfies a jump condition on the contour Γ with the jump matrix V(z,N). Assume that V(z,N) has an asymptotic expansion, as N → ∞, on Γ. An elementary proof is given for the existence of a similar type of asymptotic expansion for the matrix solution R(z,N), as n → ∞, for z ∈ ℂ\Γ. Our method makes use of only complex analysis.