Padé approximations appear to be a powerful tool to extend the validity range of expansions around certain kinematical limits and to combine expansions of different limits to a single interpolating function. After a brief outline of the general method, we will review a number of recent applications and describe the modifications that have to be applied in each case. Among these applications are...

We discuss the analytic continuation beyond μ/T ≃ 1 in QCD at nonzero T and μ by use of the Pade’ approximants. The slope of the critical line obtained in this way increases at large μ with respect to the second order Taylor result.In the hot phase Pade’ and Taylor approximants coincide, suggesting a very large, and possibly infinite, radius of convergence of the Taylor series in this thermodyn...

The scaling and squaring method is the most widely used algorithm for computing the exponential of a square matrix A. We introduce an efficient variant that uses a much smaller squaring factor when ‖A‖ 1 and a subdiagonal Padé approximant of low degree, thereby significantly reducing the overall cost and avoiding the potential instability caused by overscaling, while giving forward error of the...

In view of the successful asymptotic Padé-approximant predictions for higher-loop terms within QCD and massive scalar field theory, we address whether Padé-summations of the MS QCD β-function for a given number of flavours exhibit an infrared-stable fixed point, or alternatively, an infrared attractor of a double valued couplant as noted by Kogan and Shifman for the case of supersymmetric gluod...

For univariate functions the Kronecker theorem, stating the equivalence between the existence of an infinite block in the table of Padé approximants and the approximated function f being rational, is well-known. In [Lubi88] Lubinsky proved that if f is not rational, then its Padé table is normal almost everywhere: for an at most countable set of points the Taylor series expansion of f is such t...

Asymptotic Padé-approximant methods are utilized to estimate the O(α5s) contribution to the H → gg rate and the O(α4s) contribution to the H → bb̄ rate. The former process is of particular interest because of the slow convergence evident from the three known terms of its QCD series, which begins with an O(α2s) leading-order term. The O(α5s) contribution to the H → gg rate is expressed as a degre...

In 1961, Baker, Gammel and Wills conjectured that for functions f meromorphic in the unit ball, a subsequence of its diagonal Padé approximants converges uniformly in compact subsets of the ball omitting poles of f . There is also apparently a cruder version of the conjecture due to Padé himself, going back to the early twentieth century. We show here that for carefully chosen q on the unit cir...

Article history: Received 9 July 2010 Received in revised form 1 October 2010 Accepted 4 October 2010 Available online 12 October 2010

Periodic orbit quantization requires an analytic continuation of non-convergent semiclassical trace formulae. We propose two different methods for semiclassical quantization. The first method is based upon the harmonic inversion of semiclassical recurrence functions. A band-limited periodic orbit signal is obtained by analytical frequency windowing of the periodic orbit sum. The frequencies of ...