relation to Kummer equation

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21: 8.19 Generalized Exponential Integral

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§8.19(i) Definition and Integral Representations

… ►In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. … ►

§8.19(v) Recurrence Relation and Derivatives

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§8.19(vi) Relation to Confluent Hypergeometric Function

… ►For $U(a,b,z)$ see §13.2(i). …

22: Bibliography D

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Computational properties of three-term recurrence relations for Kummer functions. J. Comput. Appl. Math. 233 (6), pp. 1505–1510.

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ISSN 0377-0427, Document, FullText, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§13.29(iv), §13.29(iv).

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Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.

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ISSN 0962-8444, FullText, MathReview (W. Balser), ZentralBlatt

Referenced by:

§10.41(iii), §2.8(i).

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Asymptotic solutions of second-order linear differential equations having almost coalescent turning points, with an application to the incomplete gamma function. Proc. Roy. Soc. London Ser. A 452, pp. 1331–1349.

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ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§2.8(vi), §8.12, §8.12.

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

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23: 9.6 Relations to Other Functions

§9.6 Relations to Other Functions

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§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

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§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

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§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

… ►To express Airy functions in terms of hypergeometric functions combine §9.6(i) with (10.39.9).

24: Bibliography

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Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem. Comm. Pure Appl. Math. 30 (1), pp. 95–148.

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MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§23.21(ii).

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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule. Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.

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ISSN 0373-1243, MathReview, ZentralBlatt

Referenced by:

§13.29(iii).

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions. International Series of Monographs in Pure and Applied Mathematics, Vol. 66, Pergamon Press, The Macmillan Co., New York.

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, §28.1, §28.10(i), §28.11, §28.12(ii), §28.17, §28.2(ii), §28.2(iii), §28.2(iv), §28.28(i), §28.28(ii), §28.29(i), §28.32(i), §28.32(ii), §28.4(i), §28.5(i), Ch.28, §29.1, §29.11, §29.12(i), §29.12(ii), §29.12(iii), §29.14, §29.15(ii), §29.15(ii), §29.17(i), §29.18(iii), Ch.29, §30.1, §30.10, §30.11(i), §30.11(vi), §30.2(ii), §30.4(iii), §30.6, §30.8(i), §30.8(ii), Ch.30, §31.4, §31.5, §31.9(ii), Notations F, Notations G, Notations M, Notations N.

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25: 18.17 Integrals

… ►For formulas for Jacobi and Laguerre polynomials analogous to (18.17.5) and (18.17.6), see Koornwinder (1974, 1977). … ►and three formulas similar to (18.17.9)–(18.17.11) by symmetry; compare the second row in Table 18.6.1. … ►For the beta function $\mathrm{B}(a,b)$ see §5.12, and for the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. … ►Many of the Fourier transforms given in §18.17(v) have analytic continuations to Laplace transforms. … ►For the confluent hypergeometric function ${}_1{}^{}F_1^{}$ see (16.2.1) and Chapter 13. …

26: 12.14 The Function $W(a,x)$

… ►This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. … ►

§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

… ►The differential equation …

27: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

… ►Related formulas are (17.7.3), (17.8.8) and … ►

Bailey–Daum $q$-Kummer Sum

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§17.6(iii) Contiguous Relations

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Heine’s Contiguous Relations

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$q$-Differential Equation

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28: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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Numerical Grid Methods and Their Application to Schrödinger’s Equation. NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, Vol. 412, Kluwer Academic Publishers, Dordrecht.

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ISBN 0-7923-2423-4, Document, Link, MathReview (M. Znojil)

Referenced by:

§18.38(i).

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An extension of a Kummer’s quadratic transformation formula with an application. Proc. Jangjeon Math. Soc. 16 (2), pp. 229–235.

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ISSN 1598-7264, MathReview (Petr Blaschke), ZentralBlatt

Referenced by:

§16.6, §16.6.

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

Encodings:

BibTeX

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