large a or b

(0.008 seconds)

31—40 of 566 matching pages

31: Bibliography O

… ►

A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

►

A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

… ►

A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

ⓘ

External Links:: ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii).
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 (1975b) Legendre functions with both parameters large. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 175–185.

ⓘ

External Links:: FullText, MathReview (P. F. Hsieh), ZentralBlatt
Referenced by:: §14.15(v), §14.15(v).
Encodings:: BibTeX

…

32: 12.2 Differential Equations

… ►

12.2.1

\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(az^{2}+bz+c\right)w=0,

ⓘ

Defines:: $a$ : parameter (locally), $b$ : parameter (locally) and $c$ : parameter (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $z$ : complex variable
A&S Ref:: 19.1.1
Permalink:: http://dlmf.nist.gov/12.2.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.2(i), §12.2 and Ch.12

… ►Its importance is that when

a

is negative and

|a|

is large,

U\left(a,x\right)

and

\overline{U}\left(a,x\right)

asymptotically have the same envelope (modulus) and are

\tfrac{1}{2}\pi

out of phase in the oscillatory interval

-2\sqrt{-a}<x<2\sqrt{-a}

. …

33: 18.2 General Orthogonal Polynomials

… ►Assume that the interval

[a,b]

is bounded. … ►

The Nevai class ${\mathbf{M}}(a,b)$

… ►For a large class of OP’s

p_{n}

there exist pairs of differentiation formulas … ►For OP’s

p_{n}

[a,b]

with weight function

w(x)

and orthogonality relation (18.2.5_5) assume that

b<\infty

and

w(x)

is non-decreasing in the interval

[a,b]

. Then the functions

\sqrt{w(x)}p_{n}(x)

attain their maximum in

[a,b]

for

x=b

. …

34: Bibliography S

… ►

J. B. Seaborn (1991) Hypergeometric Functions and Their Applications. Texts in Applied Mathematics, Vol. 8, Springer-Verlag, New York.

ⓘ

External Links:: ISBN 0-387-97558-6, MathReview (Wolfgang Bühring), ZentralBlatt
Referenced by:: Erratum (V1.2.0) Section 18.39.
Encodings:: BibTeX

… ►

N. T. Shawagfeh (1992) The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

ⓘ

External Links:: ISSN 0253-424X, MathReview
Referenced by:: §9.10(v).
Encodings:: BibTeX

… ►

B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.

ⓘ

External Links:: Document
Referenced by:: §22.19(ii).
Encodings:: BibTeX

… ►

B. Simon (2011) Szegő’s Theorem and Its Descendants. Spectral Theory for

L^{2}

Perturbations of Orthogonal Polynomials. M. B. Porter Lectures, Princeton University Press, Princeton, NJ.

ⓘ

External Links:: ISBN 978-0-691-14704-8, MathReview (Harry Dym)
Referenced by:: §18.2(xi).
Encodings:: BibTeX

… ►

SLATEC (free Fortran library)

ⓘ

Notes:: The SLATEC Common Math Library is a large general purpose mathematical software library with broad coverage of elementary and special functions. Implementations in both single and double precision are provided. Developed by a consortium of US national laboratories. See Buzbee (1984).
External Links:: GAMS (package SLATEC)
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

35: Bibliography B

Bibliography B

… ►

L. Baker (1992) C Mathematical Function Handbook. McGraw-Hill, Inc., New York.

ⓘ

Notes:: Includes diskette containing a large collection of mathematical function software written in C; Double-precision, maximum accuracy 9S.
External Links:: ISBN 00-791-1158-0
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

… ►

C. M. Bender and T. T. Wu (1973) Anharmonic oscillator. II. A study of perturbation theory in large order. Phys. Rev. D 7, pp. 1620–1636.

ⓘ

External Links:: Document
Referenced by:: §22.19(ii).
Encodings:: BibTeX

… ►

L. C. Biedenharn, R. L. Gluckstern, M. H. Hull, and G. Breit (1955) Coulomb functions for large charges and small velocities. Phys. Rev. (2) 97 (2), pp. 542–554.

ⓘ

External Links:: Document, MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §33.5(iv).
Encodings:: BibTeX

… ►

A. A. Bogush and V. S. Otchik (1997) Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §31.13.
Encodings:: BibTeX

…

36: 2.5 Mellin Transform Methods

… ►with

a<c<b

. … ►where

a<c<b

. … ►where

J_{\nu}

denotes the Bessel function (§10.2(ii)), and

x

is a large positive parameter. … ►From (2.5.12) and (2.5.13), it is seen that

a_{s}=b_{s}=0

when

s

is even. … ►for

\Re z<b

. …

37: 3.8 Nonlinear Equations

… ►for all

n

sufficiently large, where

A

and

p

are independent of

n

, then the sequence is said to have convergence of the

p

th order. … ►

(b)

$f(x_{0})f^{\prime\prime}(x_{0})<0$ , $f^{\prime}(x)$ , $f^{\prime\prime}(x)$ do not change sign in the interval $(x_{0},x_{1})$ , and $\xi\in[x_{0},x_{1}]$ (monotonic convergence after the first iteration).

… ►If

f(a)f(b)<0

with

a<b

, then the interval

[a,b]

contains one or more zeros of

f

. …All zeros of

f

in the original interval

[a,b]

can be computed to any predetermined accuracy. …

38: 33.9 Expansions in Series of Bessel Functions

… ►

33.9.3

F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \eta)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}b_{k}t^{k/2}I_{k}\left(% \textstyle 2\sqrt{t}\right),

\eta>0

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $!$ : factorial (as in $n!$ ), $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter and $b_{k}$ : coefficient
A&S Ref:: 14.4.1
Referenced by:: §33.9(ii), §33.9(ii)
Permalink:: http://dlmf.nist.gov/33.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §33.9(ii), §33.9 and Ch.33

►

33.9.4

F_{\ell}\left(\eta,\rho\right)=C_{\ell}\left(\eta\right)\frac{(2\ell+1)!}{(2% \left|\eta\right|)^{2\ell+1}}\rho^{-\ell}\*\sum_{k=2\ell+1}^{\infty}\!\!b_{k}t% ^{k/2}J_{k}\left(\textstyle 2\sqrt{t}\right),

\eta<0

ⓘ

Symbols:: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $!$ : factorial (as in $n!$ ), $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter and $b_{k}$ : coefficient
Referenced by:: §33.9(ii), §33.9(ii)
Permalink:: http://dlmf.nist.gov/33.9.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §33.9(ii), §33.9 and Ch.33

►Here

b_{2\ell}=b_{2\ell+2}=0

b_{2\ell+1}=1

, and ►

33.9.5

{4\eta^{2}(k-2\ell)b_{k+1}+kb_{k-1}+b_{k-2}=0},

k=2\ell+2,2\ell+3,\dots

ⓘ

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $b_{k}$ : coefficient
A&S Ref:: 14.4.3 ((Early printings contained an error. All editions miss $b_{2L}=0$ .))
Permalink:: http://dlmf.nist.gov/33.9.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §33.9(ii), §33.9 and Ch.33

… ►

33.9.7

\lambda_{\ell}(\eta)\sim\sum_{k=2\ell+1}^{\infty}(-1)^{k}(k-1)!b_{k}.

ⓘ

Defines:: $\lambda_{\ell}(\eta)$ : coefficient (locally)
Symbols:: $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $b_{k}$ : coefficient
A&S Ref:: 14.4.4
Permalink:: http://dlmf.nist.gov/33.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §33.9(ii), §33.9 and Ch.33

…

39: 36.12 Uniform Approximation of Integrals

… ►where

k

is a large real parameter and

\mathbf{y}=\{y_{1},y_{2},\dots\}

is a set of additional (nonasymptotic) parameters. …

40: Ronald F. Boisvert

… ►Ronald F. Boisvert (b. …