Thomas Schmelzer

Computing the Gamma Function Using Contour Integrals and Rational Approximations

Some of the best methods for computing the gamma function are based on numerical evaluation of Hankel’s contour integral. For example, Temme evaluates this integral based on steepest descent contours by the trapezoid rule. Here we investigate a different approach to the integral: the application of the trapezoid rule on Talbot-type contours using optimal parameters recently derived by Weideman for computing inverse Laplace transforms. Relatedly, we also investigate quadrature formulas derived from best approximations to

Ricci flows connecting Taub?Bolt and Taub?NUT metrics

\exp(z)

Taming a Hydra of Singularities

on the negative real axis, following Cody, Meinardus, and Varga. The two methods are closely related, and both converge geometrically. We find that the new methods are competitive with existing ones, even though they are based on generic tools rather than on specific analysis of the gamma function.

Talbot quadratures and rational approximations

We use the Ricci flow with surgery to study four-dimensional SU(2) × U(1)-symmetric metrics on a manifold with fixed boundary given by a squashed 3-sphere. Depending on the initial metric we show that the flow converges to either the Taub–Bolt or the Taub–NUT metric, the latter case potentially requiring surgery at some point in the evolution. The Ricci flow allows us to explore the Euclidean action landscape within this symmetry class. This work extends the recent work of Headrick and Wiseman (2006 Class. Quantum Grav. 23 6683) to more interesting topologies.

Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals

We generalize a devilish integral-evaluation problem of Brian Davies that was meant to challenge Nick Trefethen and his Problem Solving Squad at Oxford. The results given in this short note are, as we think, an unexpectedly twinkling gem of classical analysis.