We develop Trapezoid, Midpoint, and Simpson’s rules for the Riemann-Stieltjes integral, the latter two being new. These rules are completely natural when the notion of relative convexity is used. Mathematics subject classification (2010): 65D30.

We prove that the functions Φ(x) = [Γ(x + 1)]1/x(1 + 1/x)x/x and log Φ(x) are Stieltjes transforms. 2000 Mathematics Subject Classification: primary 33B15; secondary 26A48

We consider the limit of empirical spectral distribution Laplace matrices generalized random graphs. Applying Stieltjes transform method, we prove under general conditions that converges to free convolution semicircular law and normal law.