F. Ursell (1980) Integrals with a large parameter: A double complex integral with four nearly coincident saddle-points. Math. Proc. Cambridge Philos. Soc. 87 (2), pp. 249–273.

ⓘ

External Links:: ISSN 0305-0041, Document, MathReview (F. I. Ibragimov), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

►

F. Ursell (1984) Integrals with a large parameter: Legendre functions of large degree and fixed order. Math. Proc. Cambridge Philos. Soc. 95 (2), pp. 367–380.

ⓘ

External Links:: ISSN 0305-0041, Document, MathReview (Roderick Wong), ZentralBlatt
Referenced by:: §14.26.
Encodings:: BibTeX

…

13: 10.21 Zeros

… ►

§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

… ► … ►

§10.21(vii) Asymptotic Expansions for Large Order

… ►

§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►The asymptotic expansion of the large positive zeros (not necessarily the

m

th) of the function …

14: Bibliography I

… ►

IMSL (commercial C, Fortran, and Java libraries) IMSL Nuerical Libraries..

ⓘ

Notes:: The IMSL Libraries are large general purpose numerical software libraries with broad coverage of elementary and special functions. Implementations are in single and double precision.
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.
Encodings:: BibTeX

…

15: 2.10 Sums and Sequences

… ►for large

n

. … ►As a first estimate for large

n

… ►(5.11.7) shows that the integrals around the large quarter circles vanish as

n\to\infty

. … ►

Example

… ►In Condition (c) we have …

16: Bibliography L

… ►

H. T. Lau (1995) A Numerical Library in C for Scientists and Engineers. CRC Press, Boca Raton, FL.

ⓘ

Notes:: With 1 IBM-PC floppy disk (3.5 inch; HD) containing a large general purpose mathematical software collection written in C including a collection of special function classes. The functions are computed for real variables only. (See Lau (2004) for Java version).
External Links:: ISBN 0-8493-7376-X, MathReview, ZentralBlatt
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, H. T. Lau (2004).
Encodings:: BibTeX

►

H. T. Lau (2004) A Numerical Library in Java for Scientists & Engineers. Chapman & Hall/CRC, Boca Raton, FL.

ⓘ

Notes:: With 1 CD-ROM (Windows, Macintosh and Unix) containing a large general purpose mathematical software collection written in Java including a collection of special function classes. The functions are computed for real variables only. (See Lau (1995) for C version).
External Links:: ISBN 1-58488-430-4, MathReview Entry, ZentralBlatt
Referenced by:: H. T. Lau (1995).
Encodings:: BibTeX

…

17: 2.3 Integrals of a Real Variable

… ►

(c)

The integral (2.3.13) converges absolutely for all sufficiently large $x$ .

…

18: 18.15 Asymptotic Approximations

… ►For large

\beta

, fixed

\alpha

, and

0\leq n/\beta\leq c

, Dunster (1999) gives asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

that are uniform in unbounded complex

z

-domains containing

z=\pm 1

. …This reference also supplies asymptotic expansions of

P^{(\alpha,\beta)}_{n}\left(z\right)

for large

n

, fixed

\alpha

, and

0\leq\beta/n\leq c

. …

19: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta\right|$

►

§33.10(i) Large $\rho$

… ►

§33.10(ii) Large Positive $\eta$

… ►

§33.10(iii) Large Negative $\eta$

… ►

C_{0}\left(\eta\right)\sim(-2\pi\eta)^{1/2}.

20: 33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

§33.5 Limiting Forms for Small $\rho$ , Small $|\eta|$ , or Large $\ell$

… ►

F_{\ell}\left(\eta,\rho\right)\sim C_{\ell}\left(\eta\right)\rho^{\ell+1},

►

F_{\ell}'\left(\eta,\rho\right)\sim(\ell+1)C_{\ell}\left(\eta\right)\rho^{\ell}.

… ►

33.5.6

C_{\ell}\left(0\right)=\frac{2^{\ell}\ell!}{(2\ell+1)!}=\frac{1}{(2\ell+1)!!}.

ⓘ

Symbols:: $C_{\NVar{\ell}}\left(\NVar{\eta}\right)$ : normalizing constant for Coulomb radial functions, $!!$ : double factorial, $!$ : factorial (as in $n!$ ) and $\ell$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/33.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §33.5(ii), §33.5 and Ch.33

… ►

§33.5(iv) Large $\ell$

…

large c

11—20 of 82 matching pages

11: Bibliography N

12: Bibliography U

13: 10.21 Zeros

§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

§10.21(vii) Asymptotic Expansions for Large Order

§10.21(viii) Uniform Asymptotic Approximations for Large Order

14: Bibliography I

15: 2.10 Sums and Sequences

Example

16: Bibliography L

17: 2.3 Integrals of a Real Variable

18: 18.15 Asymptotic Approximations

19: 33.10 Limiting Forms for Large ρ or Large | η |

§33.10 Limiting Forms for Large ρ or Large | η |

§33.10(i) Large ρ

§33.10(ii) Large Positive η

§33.10(iii) Large Negative η

20: 33.5 Limiting Forms for Small ρ , Small | η | , or Large ℓ

§33.5 Limiting Forms for Small ρ , Small | η | , or Large ℓ

§33.5(iv) Large ℓ