# Reports of Exploratory research

*Research within the* *project: **Exploratory research*

*Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions*

J.L. López, N.M. Temme

1998, MAS-R9828, ISSN 1386-3703

Bernoulli and Euler polynomials are considered for large values of the order. Convergent expansions are obtained for*B*_{n}(*nz*+1/2) and*E*(_{n}*nz*+1/2) in powers of*n*^{-1}, with coefficients being rational functions of*z*and hyperbolic functions of argument 1/2*z*. These expansions are uniformly valid for |*z*±*i*/2π| > 1/2π and |*z*±*i*/π| > 1/π, respectively. For real argument, accuracy of these approximations is restricted to the monotonic region. The range of validity of the uniformity parameter*z*is enlarged, respectively, to regions of the form |*z*±*i*/2(*m*+1)π | > 1/2(*m*+1)π and |*z*±*i*/(2*m*+1)π | > 1/(2*m*+1)π,*m*=1,2,3,.., by adding certain combinations of incomplete gamma functions to those uniform expansions. In addition, the convergence of these improved expansions is stronger and also for real argument the accuracy of these improved approximations is better in the oscillation region.

Download the pdf-file*Recent problems from uniform asymptotic analysis of integrals, in particular in connection with Tricomi's Psi-function*

N.M. Temme

1998, MAS-R9802, ISSN 1386-3703

The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's Ψ-function, in particular we consider an expansion of Tricomi-Carlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions.

Download the pdf-file*Analytical methods for a selection of elliptic singular perturbation problems*N.M. Temme

1997, MAS-R9727, ISSN 1386-3703

We consider several model problems from a class of elliptic perturbation equations in two dimensions. The domains, the differential operators, the boundary conditions, and so on, are rather simple, and are chosen in a way that the solutions can be obtained in the form of integrals or Fourier series. By using several techniques from asymptotic analysis (saddle point methods, for instance) we try to construct asymptotic approximations with respect to the small parameter that multiplies the differential operator of highest order. In particular we consider approximations that hold uniformly in the so-called boundary layers. We also pay attention to how to obtain a few terms in the asymptotic expansion by using direct methods based on singular perturbation methods.

Download the pdf-file*Asymptotics and zero distribution of Pade polynomials associated with the exponential function*

K.A. Driver, N.M. Temme

1997, MAS-R9726, ISSN 1386-3703

The polynomials*P*and_{n}*Q*having degrees_{m}*n*and*m*respectively, with*P*monic, that solve the approximation problem_{n}*P*(_{n}*z*)*e*(^{-z}+Q_{m}*z*)=(**O***z*)^{n+m+1}will be investigated for their asymptotic behavior, in particular in connection with the distribution of their zeros. The symbol

means that the left-hand side should vanish at the origin at least to the order**O***n+m+*1. This problem is discussed in great detail in a series of papers by Saff and Varga. In the present paper we show how their results can be obtained by using uniform expansions of integrals in which Airy functions are the main approximants. We give approximations of the zeros of*P*and_{n}*Q*in terms of zeros of certain Airy functions, as well of those of the remainder defined by_{m}*E*(_{n,m}*z*)=*P*(_{n}*z*)*e*(^{-z}+Q_{m}*z*).

Download the pdf-file*On polynomials related with Hermite-Pade approximations to the exponential function*K.A. Driver, N.M. Temme

1997, MAS-R9716, ISSN 1386-3703

We investigate the polynomials*P*and_{n }, Q_{m}*R*, having degrees_{s}*n, m*and*s*respectively, with*P*monic, that solve the approximation problem_{n}*E*(_{nms}*x*):=*P*(_{n}*x*)*e*(^{-2x}+ Q_{m}*x*)*e*(^{-x}+R_{s}*x*)=*O*(*x*) as^{n+m+s+2}*x*→ 0.We give a connection between the coefficients of each of the polynomials

*P*and_{n}, Q_{m}*R*and certain hypergeometric functions, which leads to a simple expression for_{s}*Q*in the case_{m}*n=s*. The approximate location of the zeros of*Q*, when_{m}*n>> m*and*n=s*, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations of*P*and_{n }, Q_{m }, R_{s}*E*are given and, using saddle point methods, we derive the exact asymptotics of_{nms}*P*and_{n }, Q_{m}*R*as_{s}*n,m*and*s*tend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomials*P*and_{n }, Q_{m}*R*for the case_{s}*n=s=*40,*m=*45.

Download the pdf-file*Numerical algorithms for uniform Airy-type asymptotic expansions*N.M. Temme

1997, MAS-R9706, ISSN 1386-3703

Airy-type asymptotic representations of a class of special functions are considered from a numerical point of view. It is well known that the evaluation of the coefficients of the asymptotic series near the transition point is a difficult problem. We discuss two methods for computing the asymptotic series. One method is based on expanding the coefficients of the asymptotic series in Maclaurin series. In the second method we consider auxiliary functions that can be computed more efficiently than the coefficients in the first method, and we don't need the tabulation of many coefficients. The methods are quite general, but the paper concentrates on Bessel functions, in particular on the differential equation of the Bessel functions, which has a turning point character when order and argument of the Bessel functions are equal.

Download the pdf-file*Zero and pole distribution of diagonal Pade approximants to the exponential function*

K.A. Driver, N.M. Temme

1997, MAS-R9701, ISSN 1386-3703

The polynomials*P*and_{n}*Q*having degrees_{m}*n*and*m*respectively, with*P*monic, that solve the approximation problem_{n}*P*(_{n}*z*)*e*(^{-z}+Q_{m}*z*)=(**O***z*)^{n+m+1}will be investigated for their asymptotic behaviour, in particular in connection with the distribution of their zeros. The symbol

means that the left-hand side should vanish at the origin at least to the order**O***n+m+*1. This problem is discussed in great detail in a series of papers by Saff and Varga. In the present paper we show how their results can be obtained by using uniform expansions of integrals in which Airy functions are the main approximants. We shall focus on the important diagonal case when*n=m*and the polynomials*P*and_{n}*Q*as well as the remainder_{n}*E*(_{n,n}*z*)=*P*(_{n}*z*)*e*(^{-z}+Q_{n}*z*) can be expressed in terms of Hankel and Bessel functions. The approximate location of the zeros of*P*and_{n}, Q_{n}*E*in terms of the known zeros of certain Airy functions. An application is given in which the asymptotic information on the zeros is used to obtain an estimate in an approximation of the unit block function by means of the polynomials_{n,n}*P*._{n}, Q_{n}

Download the pdf-file*Discrete approximations for singularly perturbed boundary value problems protect with parabolic layers*

Paul Farrell, P.W. Hemker, G.I. Shishkin

1995, NM-R9502, ISSN 0169-0388

Singularly perturbed boundary value problems for equations of elliptic and parabolic type are studied. For small values of the perturbation parameter, parabolic boundary layers appear in these problems. If classical discretisation methods are used, the solution of the finite difference scheme and the approximation of the diffusive flux derived from it do not converge uniformly with respect to this parameter. In particular, the relative error of the diffusive flux becomes unbounded as the perturbation parameter tends to zero. Using the method of special condensing grids, we can construct difference schemes that allow approximation of the solution and the normalised diffusive flux uniformly with respect to the small parameter.

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