Bruno Gabutti

Some expansions related to the Hubbell rectangular-source integral

Abstract Four series expansions are presented and examined for the Hubbell rectangular-source integral f(a, b) = ∫b0 arctan (A/ 1 + x 2 )( 1 + x 2 ) −1 dx for the extremal situations when one or both of the parameters a, b tend to zero or infinity.

Some generalizations of the Euler-Knopp transformation

SummaryThe purpose of this paper is to construct a generalization of the Euler-Knopp transformation. Using this, one may recover previously known transformations, derive new transformations useful for numerical calculations and derive generating functions and other formulas of theoretical interest involving well-known functions.

On two methods for accelerating convergence of series

SummaryThe Euler-Knopp transformation and a recently considered transformation, effective for entire function of order 1, are applied to series involving completely monotonic coefficients. Some properties of the resulting series are analyzed; these include uniform convergence with respect to the index, “a priori” and “a posteriori” estimates of the remainder. For the latter transformation a compact recursive algorithm is established which enables one to make effective use of the transformation. To illustrate the effectiveness of the transformations three applications, with examples, are included.

Uniform approximation on unit disk of series arising in plate contact problems

Slowly convergent series on the unit disk are reconsidered. Typical cases arise in plate contact problems. Uniformly converging sequences of approximants, which are expressible by recurrence relations, are established. Accurate error estimates, which prove to be realistic, are also derived. Several numerical examples confirm the theoretical predictions and show clearly the effectiveness of the method.

New Asymptotics for the Zeros of Whittaker's Functions

Uniform asymptotic representations for the zeros of the Whittaker functions Mκ,μ(z) and Wκ,μ(z) are derived from well-known uniform asymptotic expansions. The approximation formulas involve the zeros of Bessel or Airy functions.

The numerical performance of Tricomi's formula for inverting the Laplace transform

SummaryA method for inverting the Laplace transform which uses an expansion into Laguerre polynomials is considered. By means of a recently established generalization of the Euler-Knopp transformation the rate of convergence of the series of Laguerre polynomials is accelerated. For computing the transformed series a recursive algorithm is given. Results of theoretical and practical nature make the usefulness of the new transformation evident.

Evaluation of q-gamma function and q-analogues by iterative algorithms

Two known two-dimensional algorithms, obtained by modifying the classical arithmetic-harmonic mean, are reconsidered. Some rapidly convergent sequences associated with the algorithms are established and applied to the evaluation of q-analogous functions. Computation of q-gamma function, q-beta function, and q-exponential function is shown to be effective.

A novel approach for the determination of asymptotic expansions of certain oscillatory integrals

Abstract A standard method for deriving asymptotic expansion consists of applying integration by parts to standard integral representation. In this paper we consider cases in which this method is ineffective. To deal with these, we derive alternative integral representations which are amenable to this and other standard methods.

An algorithm for computing generalized Euler's transformations of series

An algorithm for computing generalized Eulers transformations of series is established. In comparison with known algorithm, the number of the arithmetic operations results to be slightly reduced. Numerical comparisons with the ε-algorithm are displayed.ZusammenfassungEin Algorithmus für die Berechnung von verallgemeinerten Euler-Transformationen wird vorgelegt. Im Vergleich mit bekannten Algorithmen wurde die Zahl der arithmetischen Operationen etwas vermindert. Numerische Vergleiche mit dem ε-Algorithmus wurden durchgeführt.

Numerical inversion of the Mellin transform by accelerated series of Laguerre polynomials

Abstract A method for inverting the Mellin transform which uses an expansion in Laguerre polynomials is considered. A procedure and related algorithm for accelerating the rate of convergence of the series is illustrated. The effectiveness of the new method is shown by means of results of theoretical and practical interest.