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1 Preface xiii 1 Introduction 1 I Basic Methods 13 2 Convergent and Divergent Series Introduction Power series: First steps Further practical aspects Differential equations and Frobenius series solutions Singular points The solution near a regular point Power series expansions around a regular singular point The Liouville transformation Hypergeometric series The Gauss hypergeometric function Other power series for the Gauss hypergeometric function Removable singularities Asymptotic expansions Watson s lemma Estimating the remainders of asymptotic expansions Exponentially improved asymptotic expansions Alternatives of asymptotic expansions Chebyshev Expansions Introduction Basic results on interpolation The Runge phenomenon and the Chebyshev nodes Chebyshev polynomials: Basic properties Properties of the Chebyshev polynomials T n (x) Chebyshev polynomials of the second, third, and fourth kinds vii

2 viii 3.4 Chebyshev interpolation Computing the Chebyshev interpolation polynomial Expansions in terms of Chebyshev polynomials Convergence properties of Chebyshev expansions Computing the coefficients of a Chebyshev expansion Clenshaw s method for solutions of linear differential equations with polynomial coefficients Evaluation of a Chebyshev sum Clenshaw s method for the evaluation of a Chebyshev sum Economization of power series Example: Computation of Airy functions of real variable Chebyshev expansions with coefficients in terms of special functions Linear Recurrence Relations and Associated Continued Fractions Introduction Condition of three-term recurrence relations Minimal solutions Perron s theorem Scaled recurrence relations Minimal solutions of TTRRs and continued fractions Some notable recurrence relations The confluent hypergeometric family The Gauss hypergeometric family Computing the minimal solution of a TTRR Miller s algorithm when a function value is known Miller s algorithm with a normalizing sum Anti-Miller algorithm Inhomogeneous linear difference equations Inhomogeneous first order difference equations. Examples Inhomogeneous second order difference equations Olver s method Anomalous behavior of some second order homogeneous and first order inhomogeneous recurrences A canonical example: Modified Bessel function Other examples: Hypergeometric recursions A first order inhomogeneous equation A warning Quadrature Methods Introduction Newton Cotes quadrature: The trapezoidal and Simpson s rule The compound trapezoidal rule The recurrent trapezoidal rule Euler s summation formula and the trapezoidal rule Gauss quadrature...132

3 ix Basics of the theory of orthogonal polynomials and Gauss quadrature The Golub Welsch algorithm Example: The Airy function in the complex plane Further practical aspects of Gauss quadrature The trapezoidal rule on R Contour integral formulas for the truncation errors Transforming the variable of integration Contour integrals and the saddle point method The saddle point method Other integration contours Integrating along the saddle point contours and examples 165 II Further Tools and Methods Numerical Aspects of Continued Fractions Introduction Definitions and notation Equivalence transformations and contractions Special forms of continued fractions Stieltjes fractions Jacobi fractions Relation with Padé approximants Convergence of continued fractions Numerical evaluation of continued fractions Steed s algorithm The modified Lentz algorithm Special functions and continued fractions Incomplete gamma function Gauss hypergeometric functions Computation of the Zeros of Special Functions Introduction Some classical methods The bisection method The fixed point method and the Newton Raphson method Complex zeros Local strategies: Asymptotic and other approximations Asymptotic approximations for large zeros Other approximations Global strategies I: Matrix methods The eigenvalue problem for orthogonal polynomials The eigenvalue problem for minimal solutions of TTRRs Global strategies II: Global fixed point methods Zeros of Bessel functions...213

4 x The general case Asymptotic methods: Further examples Airy functions Scorer functions The error functions The parabolic cylinder function Bessel functions Orthogonal polynomials Uniform Asymptotic Expansions Asymptotic expansions for the incomplete gamma functions Uniform asymptotic expansions Uniform asymptotic expansions for the incomplete gamma functions The uniform expansion Expansions for the coefficients Numerical algorithm for small values of η A simpler uniform expansion Airy-type expansions for Bessel functions The Airy-type asymptotic expansions Representations of a s (ζ), b s (ζ), c s (ζ), d s (ζ) Properties of the functions A ν,b ν,c ν,d ν Expansions for a s (ζ), b s (ζ), c s (ζ), d s (ζ) Evaluation of the functions A ν (ζ), B ν (ζ) by iteration Airy-type asymptotic expansions obtained from integrals Airy-type asymptotic expansions How to compute the coefficients α n,β n Application to parabolic cylinder functions Other Methods Introduction Padé approximations Padé approximants and continued fractions How to compute the Padé approximants Padé approximants to the exponential function Analytic forms of Padé approximations Sequence transformations The principles of sequence transformations Examples of sequence transformations The transformation of power series Numerical examples Best rational approximations Numerical solution of ordinary differential equations: Taylor expansion method Taylor-series method: Initial value problems Taylor-series method: Boundary value problem Other quadrature methods...294

5 xi Romberg quadrature Fejér and Clenshaw Curtis quadratures Other Gaussian quadratures Oscillatory integrals III Related Topics and Examples Inversion of Cumulative Distribution Functions Introduction Asymptotic inversion of the complementary error function Asymptotic inversion of incomplete gamma functions The asymptotic inversion method Determination of the coefficients ε i Expansions of the coefficients ε i Numerical examples Generalizations Asymptotic inversion of the incomplete beta function The nearly symmetric case The general error function case The incomplete gamma function case Numerical aspects High order Newton-like methods Further Examples Introduction The Euler summation formula Approximations of Stirling numbers Definitions Asymptotics for Stirling numbers of the second kind Stirling numbers of the first kind Symmetric elliptic integrals The standard forms in terms of symmetric integrals An algorithm Other elliptic integrals Numerical inversion of Laplace transforms Complex Gauss quadrature Deforming the contour Using Padé approximations IV Software Associated Algorithms Introduction Errors and stability: Basic terminology...356

6 xii Design and testing of software for computing functions: General philosophy Scaling the functions Airy and Scorer functions of complex arguments Purpose Algorithms Associated Legendre functions of integer and half-integer degrees Purpose Algorithms Bessel functions Modified Bessel functions of integer and half-integer orders Modified Bessel functions of purely imaginary orders Parabolic cylinder functions Purpose Algorithm Zeros of Bessel functions Purpose Algorithm List of Algorithms 387 Bibliography 389 Index 405

Index. for Ɣ(a, z), 39. convergent asymptotic representation, 46 converging factor, 40 exponentially improved, 39

Index. for Ɣ(a, z), 39. convergent asymptotic representation, 46 converging factor, 40 exponentially improved, 39 Index Abramowitz function computed by Clenshaw s method, 74 absolute error, 356 Airy function contour integral for, 166 Airy functions algorithm, 359 asymptotic estimate of, 18 asymptotic expansions, 81,

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