DLMF: Untitled Document

… ►

§11.6(i) Large $|z|$ , Fixed $\nu$

… ►For the corresponding expansions for

\mathbf{H}_{\nu}\left(z\right)

and

\mathbf{L}_{\nu}\left(z\right)

combine (11.6.1), (11.6.2) with (11.2.5), (11.2.6), (10.17.4), and (10.40.1). … ►

§11.6(ii) Large $|\nu|$ , Fixed $z$

►

11.6.5

\mathbf{H}_{\nu}\left(z\right),\mathbf{L}_{\nu}\left(z\right)\sim\frac{z}{\pi% \nu\sqrt{2}}\left(\frac{ez}{2\nu}\right)^{\nu},

|\operatorname{ph}\nu|\leq\pi-\delta

.

ⓘ

Symbols:: $\mathbf{H}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Struve function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathbf{L}_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Struve function, $\operatorname{ph}$ : phase, $z$ : complex variable, $\nu$ : real or complex order and $\delta$ : arbitrary small positive constant
Referenced by:: (11.11.7)
Permalink:: http://dlmf.nist.gov/11.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §11.6(ii), §11.6 and Ch.11

… ►

§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$

…

… ►

H. A. Yamani and L. Fishman (1975)

J

-matrix method: Extensions to arbitrary angular momentum and to Coulomb scattering. J. Math. Phys. 16, pp. 410–420.

ⓘ

Referenced by:: §18.39(iv).
Encodings:: BibTeX

►

H. A. Yamani and W. P. Reinhardt (1975)

L

-squared discretizations of the continuum: Radial kinetic energy and the Coulomb Hamiltonian. Phys. Rev. A 11 (4), pp. 1144–1156.

ⓘ

Referenced by:: §18.39(iv), §18.39(iv), §18.39(iv), §18.40(ii).
Encodings:: BibTeX

… ►

T. Yoshida (1995) Computation of Kummer functions

{U}(a,b,x)

for large argument

x

by using the

\tau

-method. Trans. Inform. Process. Soc. Japan 36 (10), pp. 2335–2342 (Japanese).

ⓘ

Notes:: Japanese with English summary. Double-precision Fortran.
External Links:: ISSN 0387-5806, FullText, MathReview
Referenced by:: §13.32(ii).
Encodings:: BibTeX

►

F. L. Yost, J. A. Wheeler, and G. Breit (1936) Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).
Encodings:: BibTeX

…

… ►

E. L. Wachspress (2000) Evaluating elliptic functions and their inverses. Comput. Math. Appl. 39 (3-4), pp. 131–136.

ⓘ

External Links:: ISSN 0898-1221, Document, MathReview, ZentralBlatt
Referenced by:: §22.20(ii), §22.20(v).
Encodings:: BibTeX

… ►

P. L. Walker (1991) Infinitely differentiable generalized logarithmic and exponential functions. Math. Comp. 57 (196), pp. 723–733.

ⓘ

External Links:: ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §4.12.
Encodings:: BibTeX

… ►

M. I. Weinstein and J. B. Keller (1985) Hill’s equation with a large potential. SIAM J. Appl. Math. 45 (2), pp. 200–214.

ⓘ

External Links:: ISSN 0036-1399, Document, MathReview (H. Hochstadt), ZentralBlatt
Referenced by:: §29.7(ii).
Encodings:: BibTeX

… ►

J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

ⓘ

External Links:: Document, ZentralBlatt
Referenced by:: §33.5(ii).
Encodings:: BibTeX

… ►

R. Wong (1973a) An asymptotic expansion of

{W}_{k,\,m}(z)

with large variable and parameters. Math. Comp. 27 (122), pp. 429–436.

ⓘ

External Links:: FullText, Document, MathReview (J. A. Donaldson), ZentralBlatt
Referenced by:: §13.19.
Encodings:: BibTeX

…

… ►

§12.14(viii) Asymptotic Expansions for Large Variable

… ►

§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

… ►In the following expansions, obtained from Olver (1959),

\mu

is large and positive, and

\delta

is again an arbitrary small positive constant. … ►

Airy-type Uniform Expansions

… ► …

… ►

N. M. Temme (1986) Laguerre polynomials: Asymptotics for large degree. Technical report Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.

ⓘ

External Links:: FullText
Referenced by:: §2.4(vi).
Encodings:: BibTeX

►

N. M. Temme (1987) On the computation of the incomplete gamma functions for large values of the parameters. In Algorithms for approximation (Shrivenham, 1985), Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.

ⓘ

External Links:: FullText, MathReview (G. Meinardus), ZentralBlatt
Referenced by:: §8.25(iii).
Encodings:: BibTeX

… ►

N. M. Temme (2003) Large parameter cases of the Gauss hypergeometric function. J. Comput. Appl. Math. 153 (1-2), pp. 441–462.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (J. P. Singhal), ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

… ►

N. M. Temme (2022) Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.

ⓘ

External Links:: ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §13.8(iv).
Encodings:: BibTeX

… ►

J. Todd (1954) Evaluation of the exponential integral for large complex arguments. J. Research Nat. Bur. Standards 52, pp. 313–317.

ⓘ

External Links:: MathReview (H. Bückner), ZentralBlatt
Referenced by:: §6.18(i).
Encodings:: BibTeX

…

… ► L. … L. … L. …Jenkins, L. … L. …

… ►

E. L. Ince (1926) Ordinary Differential Equations. Longmans, Green and Co., London.

ⓘ

Notes:: Reprinted by Dover Publications, New York, 1944, 1956.
External Links:: MathReview, ZentralBlatt
Referenced by:: §1.13(i), §1.13(i), §1.13(i), §31.12, §31.14(i), §32.2(i), §32.2(vi), Erratum (V1.0.25) for Section 1.13.
Encodings:: BibTeX

►

E. L. Ince (1932) Tables of the elliptic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 52, pp. 355–433.

ⓘ

External Links:: ZentralBlatt
Referenced by:: 4th item, 2nd item.
Encodings:: BibTeX

►

E. L. Ince (1940a) The periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 47–63.

ⓘ

External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.15(ii), 1st item, §29.7(i).
Encodings:: BibTeX

►

E. L. Ince (1940b) Further investigations into the periodic Lamé functions. Proc. Roy. Soc. Edinburgh 60, pp. 83–99.

ⓘ

External Links:: MathReview (H. Bateman), ZentralBlatt
Referenced by:: §29.1, §29.17(ii), §29.3(iii), §29.6(i), §29.6(ii), §29.6(iii), §29.6(iv), Ch.29, Notations E, Notations E, Notations E, Notations E.
Encodings:: BibTeX

►

L. Infeld and T. E. Hull (1951) The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.

ⓘ

Referenced by:: §18.38(iii).
Encodings:: BibTeX

…

… ►where

\mathrm{L}^{2}

is the (squared) angular momentum operator (14.30.12). The eigenfunctions of

\mathrm{L}^{2}

are the spherical harmonics

Y_{{l},{m_{l}}}\left(\theta,\phi\right)

with eigenvalues

\hbar^{2}l(l+1)

, each with degeneracy

2l+1

as

m_{l}=-l,-l+1,\dots,l

. … ►The functions

\psi_{p,l}(r)

satisfy the equation, … ►The radial Coulomb wave functions

R_{n,l}(r)

, solutions of … ►These, taken together with the infinite sets of bound states for each

l

, form complete sets. …

… ►

L. Habsieger (1988) Une

q

-intégrale de Selberg et Askey. SIAM J. Math. Anal. 19 (6), pp. 1475–1489.

ⓘ

External Links:: ISSN 0036-1410, Document, MathReview (R. A. Askey), ZentralBlatt
Referenced by:: §17.13.
Encodings:: BibTeX

… ►

B. A. Hargrave and B. D. Sleeman (1977) Lamé polynomials of large order. SIAM J. Math. Anal. 8 (5), pp. 800–842.

ⓘ

External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.16.
Encodings:: BibTeX

… ►

L. E. Hoisington and G. Breit (1938) Calculation of Coulomb wave functions for high energies. Phys. Rev. 54 (8), pp. 627–628.

ⓘ

External Links:: Document
Referenced by:: §33.7.
Encodings:: BibTeX

… ►

F. T. Howard (1996a) Explicit formulas for degenerate Bernoulli numbers. Discrete Math. 162 (1-3), pp. 175–185.

ⓘ

External Links:: ISSN 0012-365X, Document, MathReview (L. Skula), ZentralBlatt
Referenced by:: §24.16(i).
Encodings:: BibTeX

… ►

C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.

ⓘ

External Links:: ISSN 1364-5021, Document, MathReview (A. D. Wood), ZentralBlatt
Referenced by:: §10.20(ii).
Encodings:: BibTeX

…

… ►

1.8.8

L_{n}=\frac{1}{\pi}\int^{\pi}_{0}\frac{\left|\sin\left(n+\frac{1}{2}\right)t% \right|}{\sin\left(\frac{1}{2}t\right)}\,\mathrm{d}t,

n=0,1,\dots

.

ⓘ

Defines:: $L_{n}$ : Lebesgue constants (locally)
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : nonnegative integer and $\left|\NVar{x}\right|$ : absolute value of $\NVar{x}$
Permalink:: http://dlmf.nist.gov/1.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►

1.8.9

L_{n}\sim(4/{\pi}^{2})\ln n;

ⓘ

Symbols:: $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : nonnegative integer and $L_{n}$ : Lebesgue constants
Permalink:: http://dlmf.nist.gov/1.8.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §1.8(i), §1.8(i), §1.8 and Ch.1

… ►(1.8.10) continues to apply if either

a

or

b

or both are infinite and/or

f(x)

has finitely many singularities in

(a,b)

, provided that the integral converges uniformly (§1.5(iv)) at

a, b

, and the singularities for all sufficiently large

\lambda

. …

large ell

11—20 of 36 matching pages

11: 11.6 Asymptotic Expansions

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$

12: Bibliography Y

13: Bibliography W

14: 12.14 The Function $W\left(a,x\right)$

§12.14(viii) Asymptotic Expansions for Large Variable

§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

Airy-type Uniform Expansions

15: Bibliography T

16: Preface

17: Bibliography I

18: 18.39 Applications in the Physical Sciences

19: Bibliography H

20: 1.8 Fourier Series

large ell

11—20 of 36 matching pages

11: 11.6 Asymptotic Expansions

§11.6(i) Large | z | , Fixed ν

§11.6(ii) Large | ν | , Fixed z

§11.6(iii) Large | ν | , Fixed z / ν

12: Bibliography Y

13: Bibliography W

14: 12.14 The Function W ⁡ ( a , x )

§12.14(viii) Asymptotic Expansions for Large Variable

§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

Airy-type Uniform Expansions

15: Bibliography T

16: Preface

17: Bibliography I

18: 18.39 Applications in the Physical Sciences

19: Bibliography H

20: 1.8 Fourier Series

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

§11.6(iii) Large $|\nu|$ , Fixed $z/\nu$

14: 12.14 The Function $W\left(a,x\right)$