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21: Bibliography L

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On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.21(vi).

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§4.13, 2nd item, §4.48(iv).

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An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.

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Notes:

Corrigendum, ibid. 15(3) (1989), 287

External Links:

ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

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An asymptotic estimate for the Bernoulli and Euler numbers. Canad. Math. Bull. 20 (1), pp. 109–111.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.11.

Encodings:

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Two index laws for fractional integrals and derivatives. J. Austral. Math. Soc. 14, pp. 385–410.

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External Links:

Document, MathReview (C. Fox), ZentralBlatt

Referenced by:

§1.15(vi), §1.15(vii).

Encodings:

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22: Bibliography I

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The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§3.8(iii).

Encodings:

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The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

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External Links:

MathReview (D. H. Lehmer), ZentralBlatt

Referenced by:

§24.12(i).

Encodings:

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23: 18.39 Applications in the Physical Sciences

… ►Here the term $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{{\partial }^2}{{\partial x}^2}$ represents the quantum kinetic energy of a single particle of mass $m$, and $V(x)$ its potential energy. … ►

18.39.8 $$\mathscr{H}{\psi }_n(x)=\left(-\frac{{\mathrm{\hslash }}^2}{2m}\frac{d^2}{{dx}^2}+V(x)\right){\psi }_n(x)={ϵ}_n{\psi }_n(x),$$ $n=0,1,2,\mathrm{\dots },$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.39.17), (18.39.18), §18.39(i), §18.39(i), §18.39(i), §18.39(ii)

Permalink:

http://dlmf.nist.gov/18.39.E8

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See also:

Annotations for §18.39(i), §18.39(i), §18.39 and Ch.18

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18.39.38 $${𝐋}_p^m(\rho )=\frac{d^m}{{d\rho }^m}{𝐋}_p^0(\rho ),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $m$: nonnegative integer

Permalink:

http://dlmf.nist.gov/18.39.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

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18.39.39 $${𝐋}_p^0(\rho )={\mathrm{e}}^{\rho }\frac{d^p}{{d\rho }^p}({\rho }^p{\mathrm{e}}^{-\rho }),$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ and $\mathrm{e}$: base of natural logarithm

Permalink:

http://dlmf.nist.gov/18.39.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(ii), §18.39(ii), §18.39 and Ch.18

… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …

24: 20.10 Integrals

… ►For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193). …

25: Bibliography C

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Asymptotic estimates for generalized Stirling numbers. Analysis (Munich) 20 (1), pp. 1–13.

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External Links:

ISSN 0174-4747, MathReview (F. T. Howard), ZentralBlatt

Referenced by:

§26.8(vii).

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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External Links:

ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

Encodings:

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Molecular collisions and cusp catastrophes: Three methods for the calculation of Pearcey’s integral and its derivatives. Chem. Phys. Lett. 81 (2), pp. 306–310.

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External Links:

ISSN 0009-2614, Document, MathReview (J. S. Joel)

Referenced by:

§36.14(iii).

Encodings:

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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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External Links:

Document

Referenced by:

5th item.

Encodings:

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26: 20.11 Generalizations and Analogs

… ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

27: 18.40 Methods of Computation

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). ►

Derivative Rule Approach

►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that … ►Comparisons of the precisions of Lagrange and PWCF interpolations to obtain the derivatives, are shown in Figure 18.40.2. …

28: Bibliography W

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The zeros of Euler’s psi function and its derivatives. J. Math. Anal. Appl. 332 (1), pp. 607–616.

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External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt (Mehdi Hassani)

Referenced by:

§5.4(iii).

Encodings:

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:

ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

Referenced by:

§29.19(ii).

Encodings:

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Rapid approximation to the Voigt/Faddeeva function and its derivatives. J. Quant. Spect. and Rad. Transfer 62 (1), pp. 29–48.

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Notes:

Includes Fortran code in the appendices to compute the functions to an accuracy between ${10}^{-2}$ and ${10}^{-5}$.

External Links:

ISSN 0022-4073, Document

Referenced by:

§7.21, §7.21.

Encodings:

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29: 25.6 Integer Arguments

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25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

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See also:

Annotations for §25.6(i), §25.6 and Ch.25

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§25.6(ii) Derivative Values

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30: Bibliography D

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Complex zeros of linear combinations of spherical Bessel functions and their derivatives. SIAM J. Math. Anal. 4 (1), pp. 128–133.

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External Links:

ISSN 1095-7154, Document, MathReview (C. J. Bouwkamp), ZentralBlatt

Referenced by:

§10.58.

Encodings:

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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction $\zeta (s)$ de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann. Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 223–296 Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire $Mx+N$ . Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).

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Notes:

Reprinted in Collected works/Oeuvres scientifiques, Vol. I, pp. 309–425, Académie Royale de Belgique, Brussels, 2000.

External Links:

MathReview, ZentralBlatt

Referenced by:

§25.10(i), §27.11, §27.2(i).

Encodings:

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions. SIAM J. Math. Anal. 20 (3), pp. 744–760.

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Notes:

(This reference has several typographical errors.)

External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§13.21(ii), §13.21(iii), §13.21(iv).

Encodings:

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