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21: 12.14 The Function $W\left(a,x\right)$

… ►Here

w_{1}(a,x)

and

w_{2}(a,x)

are the even and odd solutions of (12.2.3): …where

\alpha_{n}(a)

and

\beta_{n}(a)

satisfy the recursion relations … ►

§12.14(vii) Relations to Other Functions

… ►The coefficients

c_{2r}

and

d_{2r}

are obtainable by equating real and imaginary parts in … ►uniformly for

t\in[-1+\delta,\infty)

, with

\zeta

\phi(\zeta)

A_{s}(\zeta)

, and

B_{s}(\zeta)

as in §12.10(vii). …

22: 1.9 Calculus of a Complex Variable

… ►If

z_{1}=x_{1}+\mathrm{i}y_{1}

z_{2}=x_{2}+\mathrm{i}y_{2}

, then … ►The series is divergent if

s_{n}

does not converge. … ►With

a_{0}=1

, …(principal value), where … ►

§1.9(vii) Inversion of Limits

…

23: 3.6 Linear Difference Equations

… ►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome. … ►Given numerical values of

w_{0}

and

w_{1}

, the solution

w_{n}

of the equation …These errors have the effect of perturbing the solution by unwanted small multiples of

w_{n}

and of an independent solution

g_{n}

, say. … ►beginning with

e_{0}=w_{0}

. … ►

§3.6(vii) Linear Difference Equations of Other Orders

…

24: 33.13 Complex Variable and Parameters

… ►The functions

F_{\ell}\left(\eta,\rho\right)

G_{\ell}\left(\eta,\rho\right)

, and

{H^{\pm}_{\ell}}\left(\eta,\rho\right)

may be extended to noninteger values of

\ell

by generalizing

(2\ell+1)!=\Gamma\left(2\ell+2\right)

, and supplementing (33.6.5) by a formula derived from (33.2.8) with

U\left(a,b,z\right)

expanded via (13.2.42). … ►The quantities

C_{\ell}\left(\eta\right)

{\sigma_{\ell}}\left(\eta\right)

, and

R_{\ell}

, given by (33.2.6), (33.2.10), and (33.4.1), respectively, must be defined consistently so that ►

33.13.1

C_{\ell}\left(\eta\right)=2^{\ell}e^{\mathrm{i}{\sigma_{\ell}}\left(\eta\right% )-(\pi\eta/2)}\Gamma\left(\ell+1-\mathrm{i}\eta\right)/\Gamma\left(2\ell+2% \right),

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, ${\sigma_{\NVar{\ell}}}\left(\NVar{\eta}\right)$ : Coulomb phase shift, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ell$ : nonnegative integer and $\eta$ : real parameter
Permalink:: http://dlmf.nist.gov/33.13.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.13 and Ch.33

… ►

33.13.2

R_{\ell}=(2\ell+1)C_{\ell}\left(\eta\right)/C_{\ell-1}\left(\eta\right).

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $\ell$ : nonnegative integer, $\eta$ : real parameter and $R_{\ell}$ : factor
Permalink:: http://dlmf.nist.gov/33.13.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §33.13 and Ch.33

…

25: 18.27 $q$ -Hahn Class

… ►In the

q

-Hahn class OP’s the role of the operator

\ifrac{\mathrm{d}}{\mathrm{d}x}

in the Jacobi, Laguerre, and Hermite cases is played by the

q

-derivative

\mathcal{D}_{q}

, as defined in (17.2.41). … ►where

X

is given by

X=\{aq^{y}\}_{y\in I_{+}}

X=\{aq^{y}\}_{y\in I_{+}}\cup\{-bq^{y}\}_{y\in I_{-}}

. Here

a, b

are fixed positive real numbers, and

I_{+}

and

I_{-}

are sequences of successive integers, finite or unbounded in one direction, or unbounded in both directions. If

I_{+}

and

I_{-}

are both nonempty, then they are both unbounded to the right. … ►

§18.27(vii) Discrete $q$ -Hermite I and II Polynomials

…

26: Bibliography O

… ►

K. Okamoto (1987a) Studies on the Painlevé equations. I. Sixth Painlevé equation

P_{{\rm VI}}

. Ann. Mat. Pura Appl. (4) 146, pp. 337–381.

ⓘ

External Links:: ISSN 0003-4622, Document, MathReview (Hélène Airault), ZentralBlatt
Referenced by:: §32.10(vi), §32.6(v), §32.7(vii), §32.7(viii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1951) A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

ⓘ

External Links:: Document, MathReview (J. Todd), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

►

F. W. J. Olver (1952) Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

ⓘ

External Links:: Document, MathReview (F. Oberhettinger), ZentralBlatt
Referenced by:: §10.19(iii), §10.21(vii).
Encodings:: BibTeX

… ►

F. W. J. Olver (1959) Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.

ⓘ

External Links:: MathReview (N. D. Kazarinoff), ZentralBlatt
Referenced by:: §12.10(i), §12.10(ii), §12.10(iii), §12.10(iv), §12.10(v), §12.10(vii), §12.10(vii), §12.11(iii), §12.11(iii), §12.14(ix), §12.14(ix), §12.14(ix), §12.14(xi), Ch.12, §18.16(v).
Encodings:: BibTeX

… ►

J. M. Ortega and W. C. Rheinboldt (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

ⓘ

Notes:: Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania
External Links:: MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))
Referenced by:: §3.8(vii).
Encodings:: BibTeX

…

27: 18.17 Integrals

… ►For the Bessel function

J_{\nu}

see §10.2(ii). … ►For the confluent hypergeometric function

{{}_{1}F_{1}}

see (16.2.1) and Chapter 13. … ►

§18.17(vii) Mellin Transforms

… ►For the hypergeometric function

{{}_{2}F_{1}}

see §§15.1 and 15.2(i). … ►For the generalized hypergeometric function

{{}_{2}F_{2}}

see (16.2.1). …

28: 16.25 Methods of Computation

… ►See §§3.6(vii), 3.7(iii), Olde Daalhuis and Olver (1998), Lozier (1980), and Wimp (1984, Chapters 7, 8).

29: Bibliography E

… ►

A. R. Edmonds (1974) Angular Momentum in Quantum Mechanics. 3rd printing, with corrections, 2nd edition, Princeton University Press, Princeton, NJ.

ⓘ

Notes:: Reprinted in 1996.
External Links:: ISBN 0-691-02589-4, MathReview, ZentralBlatt
Referenced by:: §14.30(i), §14.30(ii), §14.30(iii), §14.30(iv), §14.30(iv), §14.31(iii), (34.1.1), §34.1, §34.1, §34.13, §34.2, §34.3(i), §34.3(i), §34.3(ii), §34.3(iii), §34.3(iv), §34.3(vii), §34.3(vii), §34.4, §34.5(i), §34.5(ii), §34.5(iii), §34.5(iv), §34.5(vi), §34.6, §34.7(i), §34.7(ii), §34.7(iv), §34.7(vi), Ch.34, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.
Encodings:: BibTeX

… ►

U. T. Ehrenmark (1995) The numerical inversion of two classes of Kontorovich-Lebedev transform by direct quadrature. J. Comput. Appl. Math. 61 (1), pp. 43–72.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §10.74(vii).
Encodings:: BibTeX

… ►

Á. Elbert and A. Laforgia (1997) An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

ⓘ

External Links:: ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

… ►

G. A. Evans and J. R. Webster (1999) A comparison of some methods for the evaluation of highly oscillatory integrals. J. Comput. Appl. Math. 112 (1-2), pp. 55–69.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: §3.5(vii).
Encodings:: BibTeX

… ►

H. Exton (1983) The asymptotic behaviour of the inhomogeneous Airy function

{\rm Hi}(z)

. Math. Chronicle 12, pp. 99–104.

ⓘ

External Links:: ISSN 0581-1155, MathReview (Ram Kishore Saxena), ZentralBlatt
Referenced by:: (9.12.24), §9.12(vii).
Encodings:: BibTeX

30: 24.4 Basic Properties

… ►

24.4.28

E_{n}\left(\tfrac{1}{2}\right)=2^{-n}E_{n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer
A&S Ref:: 23.1.21
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.4.E28
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(vi), §24.4 and Ch.24

… ►

§24.4(vii) Derivatives

►

24.4.34

\frac{\mathrm{d}}{\mathrm{d}x}B_{n}\left(x\right)=nB_{n-1}\left(x\right),

n=1,2,\dots

ⓘ

Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.5
Referenced by:: §24.13(i), (25.11.7)
Permalink:: http://dlmf.nist.gov/24.4.E34
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(vii), §24.4 and Ch.24

►

24.4.35

\frac{\mathrm{d}}{\mathrm{d}x}E_{n}\left(x\right)=nE_{n-1}\left(x\right),

n=1,2,\dots

ⓘ

Symbols:: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.5
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.4.E35
Encodings:: pMML, png, TeX
See also:: Annotations for §24.4(vii), §24.4 and Ch.24

… ►Let

P(x)

denote any polynomial in

x

, and after expanding set

(B(x))^{n}=B_{n}\left(x\right)

and

(E(x))^{n}=E_{n}\left(x\right)

. …

.韩国兰芝岛世界杯公园_『网址:687.vii』篮球世界杯竞猜_b5p6v3_pr7bnb1nb.com

21—30 of 773 matching pages

21: 12.14 The Function W ⁡ ( a , x )

§12.14(vii) Relations to Other Functions

22: 1.9 Calculus of a Complex Variable

§1.9(vii) Inversion of Limits

23: 3.6 Linear Difference Equations

§3.6(vii) Linear Difference Equations of Other Orders

24: 33.13 Complex Variable and Parameters

25: 18.27 q -Hahn Class

§18.27(vii) Discrete q -Hermite I and II Polynomials

26: Bibliography O

27: 18.17 Integrals

§18.17(vii) Mellin Transforms

28: 16.25 Methods of Computation

29: Bibliography E

30: 24.4 Basic Properties

§24.4(vii) Derivatives

21: 12.14 The Function $W\left(a,x\right)$

25: 18.27 $q$ -Hahn Class

§18.27(vii) Discrete $q$ -Hermite I and II Polynomials