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Asymptotic approximations to truncation errors of series representations for special functions
 348 (SpringerVerlag
, 2007
"... Summary. Asymptotic approximations (n → ∞) to the truncation errors rn = ν=0 aν of infinite series ∑∞ ν=0 aν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation ∆rn = an+1. ..."
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Cited by 4 (4 self)
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Summary. Asymptotic approximations (n → ∞) to the truncation errors rn = ν=0 aν of infinite series ∑∞ ν=0 aν for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation ∆rn = an+1. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding EulerMaclaurin formula. In the case of the other series considered – the Gaussian hypergeometric series 2F1(a, b; c; z) and the divergent asymptotic inverse power series for the exponential integral E1(z) – the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples. 1
THE EXPONENTIALLY CONVERGENT TRAPEZOIDAL RULE
"... Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods ..."
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Abstract. It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
Vol. 45, No. 2, pp. 558–571 c ○ 2007 Society for Industrial and Applied Mathematics COMPUTING THE GAMMA FUNCTION USING CONTOUR INTEGRALS AND RATIONAL APPROXIMATIONS ∗
"... Abstract. 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 appli ..."
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Abstract. 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 Talbottype contours using optimal parameters recently derived by Weideman for computing inverse Laplace transforms. Relatedly, we also investigate quadrature formulas derived from best approximations to exp(z) 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.
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"... A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equ ..."
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A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the halfline. The new method has advantages in comparison to classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to timestep. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals. The work was supported by NSF grant ATM0620100.
Zeros of the Macdonald function of complex order
, 2006
"... The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. Th ..."
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The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. The behaviour of the zeros for varying ν  or arg ν, obtained numerically, is illustrated by means of some graphics. Key words: Macdonald function, modified Bessel function of the third kind, Hankel function, zeros
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528,
, 2006
"... The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. Th ..."
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The zzeros of the modified Bessel function of the third kind Kν(z), also known as modified Hankel function or Macdonald function, are considered for arbitrary complex values of the order ν. Approximate expressions for the zeros, applicable in the cases of very small or very large ν, are given. The behaviour of the zeros for varying ν  or arg ν, obtained numerically, is illustrated by means of some graphics. Key words: Macdonald function, modified Bessel function of the third kind, Hankel function, zeros