Olver, Frank W. J. Asymptotics and special functions. Reprint. (English) Zbl 0982.41018 AKP Classics. Wellesley, MA: A K Peters. xviii, 572 p. (1997). The first edition of this book has been published by Academic Press (1974, same title). There is a one-page new preface for the A K Peters edition in which new books and developments since the first edition of the book are mentioned. In the new edition a few minor errors have been corrected. For a review of the first edition see Zbl 0303.41035; a review of the Russian translation is given in Zbl 0712.41006. For a review by Jet Wimp, see SIAM Review, Vol. 17, No. 3, 569 – 575, 1975. The book has become a classical reference and standard work on asymptotics, in particular on rigorous asymptotics for the solutions of linear second order differential equations, which area is closely related with the classical functions of mathematical physics. Cited in 1 ReviewCited in 174 Documents MSC: 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions 34E05 Asymptotic expansions of solutions to ordinary differential equations 33-02 Research exposition (monographs, survey articles) pertaining to special functions Citations:Zbl 0303.41035; Zbl 0712.41006 PDF BibTeX XML Cite \textit{F. W. J. Olver}, Asymptotics and special functions. Reprint. Wellesley, MA: A K Peters (1997; Zbl 0982.41018) OpenURL Digital Library of Mathematical Functions: §10.12 Generating Function and Associated Series ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.14 Inequalities; Monotonicity ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.15 Derivatives with Respect to Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.16 Relations to Other Functions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.17(iii) Error Bounds for Real Argument and Order ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.17(i) Hankel’s Expansions ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.17(iv) Error Bounds for Complex Argument and Order ‣ §10.17 Asymptotic Expansions for Large Argument ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.19(ii) Debye’s Expansions ‣ §10.19 Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.19(ii) Debye’s Expansions ‣ §10.19 Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.20(ii) Complex Variables ‣ §10.20 Uniform Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.20(i) Real Variables ‣ §10.20 Uniform Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.20(ii) Complex Variables ‣ §10.20 Uniform Asymptotic Expansions for Large Order ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.21(i) Distribution ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros ‣ §10.21 Zeros ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.25(ii) Standard Solutions ‣ §10.25 Definitions ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.27 Connection Formulas ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.2(ii) Standard Solutions ‣ §10.2 Definitions ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.31 Power Series ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.34 Analytic Continuation ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.37 Inequalities; Monotonicity ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.40(iii) Error Bounds for Complex Argument and Order ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.40(ii) Error Bounds for Real Argument and Order ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.40(iii) Error Bounds for Complex Argument and Order ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.40(i) Hankel’s Expansions ‣ §10.40 Asymptotic Expansions for Large Argument ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.41(ii) Uniform Expansions for Real Variable ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.41(iii) Uniform Expansions for Complex Variable ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.41(iv) Double Asymptotic Properties ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.41(v) Double Asymptotic Properties (Continued) ‣ §10.41 Asymptotic Expansions for Large Order ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.42 Zeros ‣ Modified Bessel Functions ‣ Chapter 10 Bessel Functions §10.4 Connection Formulas ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.6(i) Recurrence Relations ‣ §10.6 Recurrence Relations and Derivatives ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.8 Power Series ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.9(iii) Products ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions §10.9(i) Integrals along the Real Line ‣ §10.9 Integral Representations ‣ Bessel and Hankel Functions ‣ Chapter 10 Bessel Functions Chapter 10 Bessel Functions Phase (or Argument) Principle ‣ §1.10(iv) Residue Theorem ‣ §1.10 Functions of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §11.10(i) Definitions ‣ §11.10 Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.10) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.11) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.14) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.15) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.16) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.17) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.18) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.19) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions (11.11.8) ‣ §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions §11.11(iii) Large ν , Fixed / z ν ‣ §11.11 Asymptotic Expansions of Anger–Weber Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions §11.2(iii) Numerically Satisfactory Solutions ‣ §11.2 Definitions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions §11.2(ii) Differential Equations ‣ §11.2 Definitions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions §11.2(i) Power-Series Expansions ‣ §11.2 Definitions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions §1.13(ii) Equations with a Parameter ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §1.13(i) Existence of Solutions ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §1.13(iv) Change of Variables ‣ §1.13 Differential Equations ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §11.6(i) Large | z | , Fixed ν ‣ §11.6 Asymptotic Expansions ‣ Struve and Modified Struve Functions ‣ Chapter 11 Struve and Related Functions §1.17(i) Delta Sequences ‣ §1.17 Integral and Series Representations of the Dirac Delta ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §11.9(iii) Asymptotic Expansion ‣ §11.9 Lommel Functions ‣ Related Functions ‣ Chapter 11 Struve and Related Functions Chapter 11 Struve and Related Functions §12.10(vi) Modifications of Expansions in Elementary Functions ‣ §12.10 Uniform Asymptotic Expansions for Large Parameter ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions §12.2(i) Introduction ‣ §12.2 Differential Equations ‣ Properties ‣ Chapter 12 Parabolic Cylinder Functions Chapter 12 Parabolic Cylinder Functions §13.14(ii) Analytic Continuation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.14(i) Differential Equation ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.14(v) Numerically Satisfactory Solutions ‣ §13.14 Definitions and Basic Properties ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.1 Special Notation ‣ Notation ‣ Chapter 13 Confluent Hypergeometric Functions §13.20(ii) Large μ , 0 ≤ κ ≤ ( - 1 δ ) μ ‣ §13.20 Uniform Asymptotic Approximations for Large μ ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.20(i) Large μ , Fixed κ ‣ §13.20 Uniform Asymptotic Approximations for Large μ ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.20(v) Large μ , Other Expansions ‣ §13.20 Uniform Asymptotic Approximations for Large μ ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.21(i) Large κ , Fixed μ ‣ §13.21 Uniform Asymptotic Approximations for Large κ ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.21(iv) Large κ , Other Expansions ‣ §13.21 Uniform Asymptotic Approximations for Large κ ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.22 Zeros ‣ Whittaker Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.2(i) Differential Equation ‣ §13.2 Definitions and Basic Properties ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.2(v) Numerically Satisfactory Solutions ‣ §13.2 Definitions and Basic Properties ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.7(i) Poincaré-Type Expansions ‣ §13.7 Asymptotic Expansions for Large Argument ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions §13.8(i) Large | b | , Fixed a and z ‣ §13.8 Asymptotic Approximations for Large Parameters ‣ Kummer Functions ‣ Chapter 13 Confluent Hypergeometric Functions Chapter 13 Confluent Hypergeometric Functions §14.12(ii) 1 < x < ∞ ‣ §14.12 Integral Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.15(iii) Large ν , Fixed μ ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.15(iii) Large ν , Fixed μ ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.15(iii) Large ν , Fixed μ ‣ §14.15 Uniform Asymptotic Approximations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.16(ii) Interval - 1 < x < 1 ‣ §14.16 Zeros ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.17(iii) Orthogonality Properties ‣ §14.17 Integrals ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.18(ii) Addition Theorems ‣ §14.18 Sums ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.1 Special Notation ‣ Notation ‣ Chapter 14 Legendre and Related Functions §14.20(vii) Asymptotic Approximations: Large τ , Fixed μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.20(vii) Asymptotic Approximations: Large τ , Fixed μ ‣ §14.20 Conical (or Mehler) Functions ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.21(iii) Properties ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.21(ii) Numerically Satisfactory Solutions ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.21(i) Associated Legendre Equation ‣ §14.21 Definitions and Basic Properties ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.23 Values on the Cut ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.24 Analytic Continuation ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.25 Integral Representations ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.25 Integral Representations ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.26 Uniform Asymptotic Expansions ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.28(ii) Heine’s Formula ‣ §14.28 Sums ‣ Complex Arguments ‣ Chapter 14 Legendre and Related Functions §14.2(iii) Numerically Satisfactory Solutions ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.2(ii) Associated Legendre Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.2(i) Legendre’s Equation ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.2(iv) Wronskians and Cross-Products ‣ §14.2 Differential Equations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.3(iii) Alternative Hypergeometric Representations ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.3(ii) Interval 1 < x < ∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.3(i) Interval - 1 < x < 1 ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.3(iv) Relations to Other Functions ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions Associated Legendre Function of the Second Kind ‣ §14.3(ii) Interval 1 < x < ∞ ‣ §14.3 Definitions and Hypergeometric Representations ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.7(ii) Rodrigues-Type Formulas ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.7(i) = μ 0 ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.7(iv) Generating Functions ‣ §14.7 Integer Degree and Order ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.8(iii) → x ∞ ‣ §14.8 Behavior at Singularities ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.8(i) → x 1 - or → x - 1 + ‣ §14.8 Behavior at Singularities ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.9(iii) Connections Between P ± μ ν ( x ) , P ± μ - - ν 1 ( x ) , Q ± μ ν ( x ) , Q μ - - ν 1 ( x ) ‣ §14.9 Connection Formulas ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.9(ii) Connections Between P ± μ ν ( ± x ) , Q - μ ν ( ± x ) , Q μ ν ( x ) ‣ §14.9 Connection Formulas ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.9(i) Connections Between P ± μ ν ( x ) , P ± μ - - ν 1 ( x ) , Q ± μ ν ( x ) , Q μ - - ν 1 ( x ) ‣ §14.9 Connection Formulas ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §14.9(iv) Whipple’s Formula ‣ §14.9 Connection Formulas ‣ Real Arguments ‣ Chapter 14 Legendre and Related Functions §1.4(ii) Continuity ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods Chapter 14 Legendre and Related Functions Functions of Bounded Variation ‣ §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §1.4(v) Definite Integrals ‣ §1.4 Calculus of One Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §15.10(i) Fundamental Solutions ‣ §15.10 Hypergeometric Differential Equation ‣ Properties ‣ Chapter 15 Hypergeometric Function §15.11(ii) Transformation Formulas ‣ §15.11 Riemann’s Differential Equation ‣ Properties ‣ Chapter 15 Hypergeometric Function §15.11(i) Equations with Three Singularities ‣ §15.11 Riemann’s Differential Equation ‣ Properties ‣ Chapter 15 Hypergeometric Function §15.12(iii) Other Large Parameters ‣ §15.12 Asymptotic Approximations ‣ Properties ‣ Chapter 15 Hypergeometric Function §15.1 Special Notation ‣ Notation ‣ Chapter 15 Hypergeometric Function §15.2(ii) Analytic Properties ‣ §15.2 Definitions and Analytical Properties ‣ Properties ‣ Chapter 15 Hypergeometric Function §15.8(i) Linear Transformations ‣ §15.8 Transformations of Variable ‣ Properties ‣ Chapter 15 Hypergeometric Function Chapter 15 Hypergeometric Function §16.2(v) Behavior with Respect to Parameters ‣ §16.2 Definition and Analytic Properties ‣ Generalized Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer G -Function §18.15(iii) Legendre ‣ §18.15 Asymptotic Approximations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials §18.15(v) Hermite ‣ §18.15 Asymptotic Approximations ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials §18.16(v) Hermite ‣ §18.16 Zeros ‣ Classical Orthogonal Polynomials ‣ Chapter 18 Orthogonal Polynomials §1.8(i) Definitions and Elementary Properties ‣ §1.8 Fourier Series ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §1.9(vi) Power Series ‣ §1.9 Calculus of a Complex Variable ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods §2.10(iii) Asymptotic Expansions of Entire Functions ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.10(ii) Summation by Parts ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.10(i) Euler–Maclaurin Formula ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations Example ‣ §2.10(ii) Summation by Parts ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations Example ‣ §2.10(iii) Asymptotic Expansions of Entire Functions ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations Example ‣ §2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method ‣ §2.10 Sums and Sequences ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.11(i) Numerical Use of Asymptotic Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.11(i) Numerical Use of Asymptotic Expansions ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.11(vi) Direct Numerical Transformations ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.11(vi) Direct Numerical Transformations ‣ §2.11 Remainder Terms; Stokes Phenomenon ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.1(ii) Integration and Differentiation ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.1(iii) Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.1(i) Asymptotic and Order Symbols ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.1(v) Generalized Asymptotic Expansions ‣ §2.1 Definitions and Elementary Properties ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.2 Transcendental Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations Example ‣ §2.2 Transcendental Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.3(iii) Laplace’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.3(ii) Watson’s Lemma ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.3(iii) Laplace’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.3(i) Integration by Parts ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method ‣ §2.3 Integrals of a Real Variable ‣ Areas ‣ Chapter 2 Asymptotic Approximations §24.12(i) Bernoulli Polynomials: Real Zeros ‣ §24.12 Zeros ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials Euler–Maclaurin Summation Formula ‣ §24.17(i) Summation ‣ §24.17 Mathematical Applications ‣ Applications ‣ Chapter 24 Bernoulli and Euler Polynomials §24.9 Inequalities ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials §2.4(iii) Laplace’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(ii) Inverse Laplace Transforms ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(iii) Laplace’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(i) Watson’s Lemma ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(ii) Inverse Laplace Transforms ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(i) Watson’s Lemma ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(iv) Saddle Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(iv) Saddle Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(vi) Other Coalescing Critical Points ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method ‣ §2.4 Contour Integrals ‣ Areas ‣ Chapter 2 Asymptotic Approximations §26.7(iv) Asymptotic Approximation ‣ §26.7 Set Partitions: Bell Numbers ‣ Properties ‣ Chapter 26 Combinatorial Analysis §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.7(ii) Irregular Singularities of Rank 1 ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.7(i) Regular Singularities: Fuchs–Frobenius Theory ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.7(iv) Numerically Satisfactory Solutions ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations Example ‣ §2.7(iii) Liouville–Green (WKBJ) Approximation ‣ §2.7 Differential Equations ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(iii) Case II: Simple Turning Point ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(ii) Case I: No Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(iii) Case II: Simple Turning Point ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(i) Classification of Cases ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(ii) Case I: No Transition Points ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(iv) Case III: Simple Pole ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations §2.8(iv) Case III: Simple Pole ‣ §2.8 Differential Equations with a Parameter ‣ Areas ‣ Chapter 2 Asymptotic Approximations Chapter 2 Asymptotic Approximations §3.9(vi) Applications and Further Transformations ‣ §3.9 Acceleration of Convergence ‣ Areas ‣ Chapter 3 Numerical Methods §4.13 Lambert W -Function ‣ Logarithm, Exponential, Powers ‣ Chapter 4 Elementary Functions §5.11(iii) Ratios ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function §5.11(i) Poincaré-Type Expansions ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function §5.11(ii) Error Bounds and Exponential Improvement ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function Terminology ‣ §5.11(i) Poincaré-Type Expansions ‣ §5.11 Asymptotic Expansions ‣ Properties ‣ Chapter 5 Gamma Function §5.12 Beta Function ‣ Properties ‣ Chapter 5 Gamma Function §5.17 Barnes’ G -Function (Double Gamma Function) ‣ Properties ‣ Chapter 5 Gamma Function §5.2(ii) Euler’s Constant ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function §5.2(i) Gamma and Psi Functions ‣ §5.2 Definitions ‣ Properties ‣ Chapter 5 Gamma Function §5.4(ii) Psi Function ‣ §5.4 Special Values and Extrema ‣ Properties ‣ Chapter 5 Gamma Function §5.4(i) Gamma Function ‣ §5.4 Special Values and Extrema ‣ Properties ‣ Chapter 5 Gamma Function §5.5(iii) Multiplication ‣ §5.5 Functional Relations ‣ Properties ‣ Chapter 5 Gamma Function §5.5(ii) Reflection ‣ §5.5 Functional Relations ‣ Properties ‣ Chapter 5 Gamma Function §5.5(i) Recurrence ‣ §5.5 Functional Relations ‣ Properties ‣ Chapter 5 Gamma Function §5.7(ii) Other Series ‣ §5.7 Series Expansions ‣ Properties ‣ Chapter 5 Gamma Function §5.8 Infinite Products ‣ Properties ‣ Chapter 5 Gamma Function §5.9(ii) Psi Function, Euler’s Constant, and Derivatives ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function §5.9(i) Gamma Function ‣ §5.9 Integral Representations ‣ Properties ‣ Chapter 5 Gamma Function Chapter 5 Gamma Function §6.12(i) Exponential and Logarithmic Integrals ‣ §6.12 Asymptotic Expansions ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §6.12(i) Exponential and Logarithmic Integrals ‣ §6.12 Asymptotic Expansions ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §6.2(ii) Sine and Cosine Integrals ‣ §6.2 Definitions and Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §6.2(i) Exponential and Logarithmic Integrals ‣ §6.2 Definitions and Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §6.4 Analytic Continuation ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §6.5 Further Interrelations ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §6.6 Power Series ‣ Properties ‣ Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals Chapter 6 Exponential, Logarithmic, Sine, and Cosine Integrals §7.12(iii) Goodwin–Staton Integral ‣ §7.12 Asymptotic Expansions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §7.12(i) Complementary Error Function ‣ §7.12 Asymptotic Expansions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §7.20(i) Asymptotics ‣ §7.20 Mathematical Applications ‣ Applications ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §7.2(iii) Fresnel Integrals ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §7.2(ii) Dawson’s Integral ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §7.2(i) Error Functions ‣ §7.2 Definitions ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §7.5 Interrelations ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals Chapter 7 Error Functions, Dawson’s and Fresnel Integrals §8.10 Inequalities ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions §8.11(i) Large z , Fixed a ‣ §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions §8.11(i) Large z , Fixed a ‣ §8.11 Asymptotic Approximations and Expansions ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions §8.2(ii) Analytic Continuation ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions §8.2(i) Definitions ‣ §8.2 Definitions and Basic Properties ‣ Incomplete Gamma Functions ‣ Chapter 8 Incomplete Gamma and Related Functions Chapter 8 Incomplete Gamma and Related Functions (9.10.11) ‣ §9.10(iv) Definite Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.10.12) ‣ §9.10(iv) Definite Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.10.17) ‣ §9.10(vi) Mellin Transform ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.10.22) ‣ §9.10(viii) Repeated Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.10(iv) Definite Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.10(viii) Repeated Integrals ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.10(vi) Mellin Transform ‣ §9.10 Integrals ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.12.10) ‣ §9.12(iv) Numerically Satisfactory Solutions ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.1) ‣ §9.12(i) Differential Equation ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.25) ‣ Functions and Derivatives ‣ §9.12(viii) Asymptotic Expansions ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.27) ‣ Functions and Derivatives ‣ §9.12(viii) Asymptotic Expansions ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.3) ‣ §9.12(i) Differential Equation ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.4) ‣ §9.12(i) Differential Equation ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.5) ‣ §9.12(i) Differential Equation ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.6) ‣ §9.12(iii) Initial Values ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.7) ‣ §9.12(iii) Initial Values ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.8) ‣ §9.12(iv) Numerically Satisfactory Solutions ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.12.9) ‣ §9.12(iv) Numerically Satisfactory Solutions ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions §9.12(iii) Initial Values ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions §9.12(i) Differential Equation ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions §9.12(viii) Asymptotic Expansions ‣ §9.12 Scorer Functions ‣ Related Functions ‣ Chapter 9 Airy and Related Functions (9.2.10) ‣ §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.11) ‣ §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.12) ‣ §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.13) ‣ §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.14) ‣ §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.15) ‣ §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.1) ‣ §9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.2) ‣ §9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.3) ‣ §9.2(ii) Initial Values ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.4) ‣ §9.2(ii) Initial Values ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.5) ‣ §9.2(ii) Initial Values ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.6) ‣ §9.2(ii) Initial Values ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.7) ‣ §9.2(iv) Wronskians ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.8) ‣ §9.2(iv) Wronskians ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.2.9) ‣ §9.2(iv) Wronskians ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.2(iii) Numerically Satisfactory Pairs of Solutions ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.2(ii) Initial Values ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.2(i) Airy’s Equation ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.2(iv) Wronskians ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions Table 9.2.1 ‣ §9.2(iii) Numerically Satisfactory Pairs of Solutions ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.2(v) Connection Formulas ‣ §9.2 Differential Equation ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.4.1) ‣ §9.4 Maclaurin Series ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.4.2) ‣ §9.4 Maclaurin Series ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.4.3) ‣ §9.4 Maclaurin Series ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.4.4) ‣ §9.4 Maclaurin Series ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.4 Maclaurin Series ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.5.1) ‣ §9.5(i) Real Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.5.2) ‣ §9.5(i) Real Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.5.4) ‣ §9.5(ii) Complex Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.5(ii) Complex Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.5(i) Real Variable ‣ §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.10) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.11) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.12) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.13) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.14) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.15) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.16) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.17) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.18) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.19) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.20) ‣ §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.2) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.3) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.4) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.5) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.6) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.7) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.8) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.6.9) ‣ §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions ‣ §9.6 Relations to Other Functions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.10) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.11) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.12) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.13) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.14) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.15) ‣ §9.7(iii) Error Bounds for Real Variables ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.2) ‣ §9.7(i) Notation ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.3) ‣ §9.7(i) Notation ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.4) ‣ §9.7(i) Notation ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.5) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.6) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.7) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.8) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.7.9) ‣ §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.7(iii) Error Bounds for Real Variables ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.7(ii) Poincaré-Type Expansions ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.7(i) Notation ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.7(iv) Error Bounds for Complex Variables ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions Table 9.7.1 ‣ §9.7(i) Notation ‣ §9.7 Asymptotic Expansions ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.14) ‣ §9.8(ii) Identities ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.18) ‣ §9.8(ii) Identities ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.1) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.2) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.3) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.4) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.5) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.6) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.7) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.8.8) ‣ §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.8(iii) Monotonicity ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.8(ii) Identities ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.8(i) Definitions ‣ §9.8 Modulus and Phase ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.9.1) ‣ §9.9(ii) Relation to Modulus and Phase ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions (9.9.2) ‣ §9.9(ii) Relation to Modulus and Phase ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.9(ii) Relation to Modulus and Phase ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions §9.9(i) Distribution and Notation ‣ §9.9 Zeros ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions Chapter 9 Airy and Related Functions Profile Frank W. J. Olver ‣ About the Project Notations P ‣ Notations Notations Q ‣ Notations Notations V ‣ Notations