Other power series for the Gauss hypergeometric function 30 2.3.3 Removable singularities 33 2.4 Asymptotic expansions 34 2.4.1 Watson's lemma 36 2.4.2 Estimating the remainders of asymptotic expansions ... 2.4.3 Exponentially improved asymptotic expansions 2.4.4 Alternatives of asymptotic expansions 3 Chebyshev Expansions 3.1 Introduction 3.2 Basic results on interpolation 3.2.1 The Runge phenomenon and the Chebyshev nodes .... 3.3 Chebyshev polynomials: Basic properties 3.3.1 Properties of the Chebyshev polynomials T n (x) 3.3.2 Chebyshev polynomials of the second, third, and fourth kinds vii viii Contents 3.4 Chebyshev interpolation 3.4.1 Computing the Chebyshev interpolation polynomial ... 3.5 Expansions in terms of Chebyshev polynomials 3.5.1 Convergence properties of Chebyshev expansions .... 3.6 Computing the coefficients of a Chebyshev expansion 3.6.1 Clenshaw's method for solutions of linear differential equations with polynomial coefficients 3.7 Evaluation of a Chebyshev sum 3.7.1 Clenshaw's method for the evaluation of a Chebyshev sum 3.8 Economization of power series 3.9 Example: Computation of Airy functions of real variable 3.10 Chebyshev expansions with coefficients in terms of special functions. 4 Linear Recurrence Relations and Associated Continued Fractions 4.1 Contents 7.5.2 The general case : 7.6 Asymptotic methods: Further examples 7.6.1 Airy functions 7.6.2 Scorer functions 7.6.3 The error functions 7.6.4 The parabolic cylinder function 7.6.5 Bessel functions 7.6.6 Orthogonal polynomials Uniform Asymptotic Expansions 8.1 Asymptotic expansions for the incomplete gamma functions 8.2 Uniform asymptotic expansions 8.3 Uniform asymptotic expansions for the incomplete gamma functions. 8.3.1 The uniform expansion 8.3.2 Expansions for the coefficients 8.3.3 Numerical algorithm for small values of JJ 8.3.4 A simpler uniform expansion 8.4 Airy-type expansions for Bessel functions 8.4.1 The Airy-type asymptotic expansions 8.4.2 Representations of a s {S),b s (S),c s (S),d s (O 8.4.3 Properties of the functions A v , B v , C v , D v 8.4.4 Expansions for a v (f)A(f).c s-(0>^(f) 8.4.5 Evaluation of the functions A v {t;), B P (£) by iteration. . .258 8.5 Airy-type asymptotic expansions obtained from integrals 8.5.1 Airy-type asymptotic expansions 8.5.2 How to compute the coefficients a n , fi n 8.5.3 Application to parabolic cylinder functions Other Methods 9.1
Taylor expansions of analytic functions are considered with respect to two points. Cauchy‐type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in two points.
Abstract. Earlier investigations on uniform asymptotic expansions of the incomplete gamma functions are reconsidered. The new results include estimations for the remainder and the extension of the results to complex variables.
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