large a (or b) and c

(0.007 seconds)

31—40 of 154 matching pages

31: 17.2 Calculus

… ►These identities are the first in a large collection of similar results. …

32: 12.2 Differential Equations

… ►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: 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

…

34: 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

…

35: Bibliography S

… ►

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

… ►

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

…

36: 2.5 Mellin Transform Methods

… ►Similarly, if

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)

and

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)

can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for

I(x)

for large values of

x

. … ►where

J_{\nu}

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

x

is a large positive parameter. …

37: 18.2 General Orthogonal Polynomials

… ►For a large class of OP’s

p_{n}

there exist pairs of differentiation formulas …

38: 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. …

39: 33.9 Expansions in Series of Bessel Functions

… ►

33.9.1

F_{\ell}\left(\eta,\rho\right)=\rho\sum_{k=0}^{\infty}a_{k}\mathsf{j}_{\ell+k}% \left(\rho\right),

ⓘ

Symbols:: $F_{\NVar{\ell}}\left(\NVar{\eta},\NVar{\rho}\right)$ : regular Coulomb radial function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $a_{k}$ : coefficients
A&S Ref:: 14.4.5
Referenced by:: §33.9(i), §33.9(i)
Permalink:: http://dlmf.nist.gov/33.9.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §33.9(i), §33.9 and Ch.33

►where the function

\mathsf{j}

is as in §10.47(ii),

a_{-1}=0

a_{0}=(2\ell+1)!!C_{\ell}\left(\eta\right)

, and ►

33.9.2

\frac{k(k+2\ell+1)}{2k+2\ell+1}a_{k}-2\eta a_{k-1}+\frac{(k-2)(k+2\ell-1)}{2k+% 2\ell-3}a_{k-2}=0,

k=1,2,\dots

ⓘ

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $a_{k}$ : coefficients
A&S Ref:: 14.4.6
Permalink:: http://dlmf.nist.gov/33.9.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §33.9(i), §33.9 and Ch.33

…

40: Ronald F. Boisvert

… ►He is also Editor at Large for the DLMF Project. ►Boisvert received a Ph. … ►He is a Fellow of the American Association for the Advancement of Science, the Association for Computing Machinery, and the Washington Academy of Sciences.