DLMF: Untitled Document

§13.3 Recurrence Relations and Derivatives

§13.3(i) Recurrence Relations

13.3.7

U\left(a-1,b,z\right)+(b-2a-z)U\left(a,b,z\right)+a(a-b+1)U\left(a+1,b,z\right% )=0,

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.15
Permalink:: http://dlmf.nist.gov/13.3.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

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13.3.14

(a+1)zU\left(a+2,b+2,z\right)+(z-b)U\left(a+1,b+1,z\right)-U\left(a,b,z\right)% =0.

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Symbols:: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.3(i)
Permalink:: http://dlmf.nist.gov/13.3.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §13.3(i), §13.3 and Ch.13

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§13.3(ii) Differentiation Formulas

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M. V. Berry and C. J. Howls (1994) Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.

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External Links:: ISSN 0962-8444, FullText, MathReview (F. Ursell), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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J. M. Blair, C. A. Edwards, and J. H. Johnson (1976) Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: §7.17(iii), §7.17(iii).
Encodings:: BibTeX

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G. Blanch and D. S. Clemm (1962) Tables Relating to the Radial Mathieu Functions. Vol. 1: Functions of the First Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 1st item.
Encodings:: BibTeX

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G. Blanch and D. S. Clemm (1965) Tables Relating to the Radial Mathieu Functions. Vol. 2: Functions of the Second Kind. U.S. Government Printing Office, Washington, D.C..

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External Links:: MathReview (C. J. Bouwkamp)
Referenced by:: 2nd item, 1st item.
Encodings:: BibTeX

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T. H. Boyer (1969) Concerning the zeros of some functions related to Bessel functions. J. Mathematical Phys. 10 (9), pp. 1729–1744.

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External Links:: Document, MathReview (C. V. Coffman), ZentralBlatt
Referenced by:: §10.21(xi).
Encodings:: BibTeX

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L. Carlitz (1963) The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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External Links:: MathReview (D. P. Banerjee), ZentralBlatt
Referenced by:: §7.17(ii).
Encodings:: BibTeX

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B. C. Carlson (2006b) 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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External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §19.25(v), §19.29(i), §22.14(ii).
Encodings:: BibTeX

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W. J. Cody (1969) Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.

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External Links:: FullText, MathReview, ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

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W. J. Cody (1991) Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:: ISSN 0098-3500, Document, MathReview, ZentralBlatt
Referenced by:: §5.24(ii), §5.24(iii).
Encodings:: BibTeX

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M. W. Coffey (2009) An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.

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External Links:: ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt
Referenced by:: §25.18(i), §25.18(i).
Encodings:: BibTeX

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G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

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C. G. van der Laan and N. M. Temme (1984) Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.

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Notes:: A reprint was published in 1986.
External Links:: ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt
Referenced by:: §5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).
Encodings:: BibTeX

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R. S. Varma (1941) An infinite series of Weber’s parabolic cylinder functions. Proc. Benares Math. Soc. (N.S.) 3, pp. 37.

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External Links:: MathReview (M. C. Gray), ZentralBlatt
Referenced by:: §12.13(ii).
Encodings:: BibTeX

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H. Volkmer (1982) Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

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External Links:: ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.8, §29.8.
Encodings:: BibTeX

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H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

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§13.15 Recurrence Relations and Derivatives

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§13.15(i) Recurrence Relations

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13.15.1

(\kappa-\mu-\tfrac{1}{2})M_{\kappa-1,\mu}\left(z\right)+(z-2\kappa)M_{\kappa,% \mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})M_{\kappa+1,\mu}\left(z\right)=0,

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
A&S Ref:: 13.4.29
Permalink:: http://dlmf.nist.gov/13.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

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13.15.2

2\mu(1+2\mu)\sqrt{z}M_{\kappa-\frac{1}{2},\mu-\frac{1}{2}}\left(z\right)-(z+2% \mu)(1+2\mu)M_{\kappa,\mu}\left(z\right)+(\kappa+\mu+\tfrac{1}{2})\sqrt{z}M_{% \kappa+\frac{1}{2},\mu+\frac{1}{2}}\left(z\right)=0,

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Symbols:: $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$ : Whittaker confluent hypergeometric function and $z$ : complex variable
Referenced by:: §13.15(i)
Permalink:: http://dlmf.nist.gov/13.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §13.15(i), §13.15 and Ch.13

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§13.15(ii) Differentiation Formulas

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Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

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Source citations

Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

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Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

The Gegenbauer function $C^{(\lambda)}_{\alpha}\left(z\right)$ , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C^{(\lambda)}_{n}\left(z\right)$ . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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Equation (7.2.3)

Originally named as a complementary error function, $w\left(z\right)$ has been renamed as the Faddeeva (or Faddeyeva) function.

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Chapter 25 Zeta and Related Functions

A number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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§9.9(ii) Relation to Modulus and Phase

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§9.9(iii) Derivatives With Respect to $k$

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§9.9(iv) Asymptotic Expansions

… тЦ║For error bounds for the asymptotic expansions of

a_{k}

,

b_{k}

,

a^{\prime}_{k}

, and

b^{\prime}_{k}

see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). тЦ║

§9.9(v) Tables

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… тЦ║As a condition of using the DLMF, you explicitly release NIST from any and all liabilities for any damage of any type that may result from errors or omissions in the DLMF. … тЦ║If you feel you have found an error in DLMF, please see Possible Errors in DLMF. … тЦ║The DLMF wishes to provide users of special functions with essential reference information related to the use and application of special functions in research, development, and education. Using special functions in applications often requires computing them. … тЦ║

Master Software Index

In association with the DLMF we will provide an index of all software for the computation of special functions covered by the DLMF. It is our intention that this will become an exhaustive list of sources of software for special functions. In each case we will maintain a single link where readers can obtain more information about the listed software. We welcome requests from software authors (or distributors) for new items to list.

Note that here we will only include software with capabilities that go beyond the computation of elementary functions in standard precisions since such software is nearly universal in scientific computing environments.

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G. Nemes (2017a) Error bounds for the asymptotic expansion of the Hurwitz zeta function. Proc. A. 473 (2203), pp. 20170363, 16.

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External Links:: ISSN 1364-5021, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.15.
Encodings:: BibTeX

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G. Nemes (2013b) Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:: ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt
Referenced by:: §5.11(ii), §5.11(ii).
Encodings:: BibTeX

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G. Nemes (2014a) Error bounds and exponential improvement for the asymptotic expansion of the Barnes

G

-function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.

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External Links:: ISSN 1364-5021, MathReview (Mehdi Hassani), ZentralBlatt
Referenced by:: §5.17, §5.17.
Encodings:: BibTeX

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G. Nemes (2015a) Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal. Proc. Roy. Soc. Edinburgh Sect. A 145 (3), pp. 571–596.

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External Links:: ISSN 0308-2105, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §5.11(i), §5.11(i).
Encodings:: BibTeX

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G. Nemes (2017b) Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions. Acta Appl. Math. 150, pp. 141–177.

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External Links:: ISSN 0167-8019, Document, MathReview Entry, ZentralBlatt
Referenced by:: (9.7.16), (9.7.17), §9.7(iii), §9.7(iv).
Encodings:: BibTeX

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… тЦ║The functions

\rho_{\nu}(t)

and

\sigma_{\nu}(t)

are related to the inverses of the phase functions

\theta_{\nu}\left(x\right)

and

\phi_{\nu}\left(x\right)

defined in §10.18(i): if

\nu\geq 0

, then … тЦ║For error bounds see Wong and Lang (1990), Wong (1995), and Elbert and Laforgia (2000). … … тЦ║An error bound is included for the case

\nu\geq\tfrac{3}{2}

. … тЦ║For error bounds for (10.21.32) see Qu and Wong (1999); for (10.21.36) and (10.21.37) see Elbert and Laforgia (1997). …

error and related functions

31—40 of 73 matching pages

31: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

§13.3(i) Recurrence Relations

§13.3(ii) Differentiation Formulas

32: Bibliography B

33: Bibliography C

34: Bibliography V

35: 13.15 Recurrence Relations and Derivatives

§13.15 Recurrence Relations and Derivatives

§13.15(i) Recurrence Relations

§13.15(ii) Differentiation Formulas

36: Errata

37: 9.9 Zeros

§9.9(ii) Relation to Modulus and Phase

§9.9(iii) Derivatives With Respect to $k$

§9.9(iv) Asymptotic Expansions

§9.9(v) Tables

38: Notices

39: Bibliography N

40: 10.21 Zeros

error and related functions

31—40 of 73 matching pages

31: 13.3 Recurrence Relations and Derivatives

§13.3 Recurrence Relations and Derivatives

§13.3(i) Recurrence Relations

§13.3(ii) Differentiation Formulas

32: Bibliography B

33: Bibliography C

34: Bibliography V

35: 13.15 Recurrence Relations and Derivatives

§13.15 Recurrence Relations and Derivatives

§13.15(i) Recurrence Relations

§13.15(ii) Differentiation Formulas

36: Errata

37: 9.9 Zeros

§9.9(ii) Relation to Modulus and Phase

§9.9(iii) Derivatives With Respect to k

§9.9(iv) Asymptotic Expansions

§9.9(v) Tables

38: Notices

39: Bibliography N

40: 10.21 Zeros

§9.9(iii) Derivatives With Respect to $k$