large c

(0.001 seconds)

21—30 of 82 matching pages

21: 2.7 Differential Equations

… ►From the numerical standpoint, however, the pair

w_{3}(z)

and

w_{4}(z)

has the drawback that severe numerical cancellation can occur with certain combinations of

C

and

D

, for example if

C

and

D

are equal, or nearly equal, and

z

, or

\Re z

, is large and negative. …

22: Preface

… ►Frechette, C. …Gebbie, C. … C. … C. … C. …

23: Bibliography G

… ►

GSL (free C library) GNU Scientific Library The GNU Project.

ⓘ

Notes:: The GNU Scientific Library is a large general purpose numerical software library with broad coverage of elementary and special functions. Implementation is in double precision.
External Links:: Home Page
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.
Encodings:: BibTeX

…

24: 8.18 Asymptotic Expansions of $I_{x}\left(a,b\right)$

… ►

§8.18(i) Large Parameters, Fixed $x$

… ►

§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

►

Large $a$ , Fixed $b$

… ►

Symmetric Case

… ►

General Case

…

25: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

►

§10.41(i) Asymptotic Forms

… ►

§10.41(ii) Uniform Expansions for Real Variable

… ► … ►

26: 2.11 Remainder Terms; Stokes Phenomenon

… ►Hence from §7.12(i)

\operatorname{erfc}\left(\sqrt{\frac{1}{2}\rho}\;c(\theta)\right)

is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when

\rho

is large. …

27: Bibliography T

… ►

W. J. Thompson (1997) Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners. John Wiley & Sons Inc., New York.

ⓘ

Notes:: With CD-ROM containing a large collection of mathematical function software written in Fortran 90 and Mathematica (an edition with the same software in C and Mathematica exists also). The functions are computed for real variables only. Maximum accuracy 12D.
External Links:: ISBN 0-471-18171-4, MathReview, ZentralBlatt
Referenced by:: §19.36(ii), § ‣ 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

…

28: 10.74 Methods of Computation

… ►If

x

|z|

is large compared with

|\nu|^{2}

, then the asymptotic expansions of §§10.17(i)–10.17(iv) are available. … ►For large positive real values of

\nu

the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large

x

|z|

, whether or not

\nu

is large. … ►And since there are no error terms they could, in theory, be used for all values of

z

; however, there may be severe cancellation when

|z|

is not large compared with

n^{2}

. … ►See Leung and Ghaderpanah (1979), Kerimov and Skorokhodov (1984b, c, 1985a, 1985b), Skorokhodov (1985), Modenov and Filonov (1986), Vrahatis et al. (1997b), and Segura (2013). …

29: 14.15 Uniform Asymptotic Approximations

… ►

§14.15(i) Large $\mu$ , Fixed $\nu$

… ►

§14.15(ii) Large $\mu$ , $0\leq\nu+\frac{1}{2}\leq(1-\delta)\mu$

… ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►

§14.15(iii) Large $\nu$ , Fixed $\mu$

… ►

30: 19.12 Asymptotic Approximations

… ►For the asymptotic behavior of

F\left(\phi,k\right)

and

E\left(\phi,k\right)

\phi\to\tfrac{1}{2}\pi-

and

k\to 1-

see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►

19.12.4

(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\frac{4}{k^{\prime}}\right% )\left(1+O\left({k^{\prime}}^{2}\right)\right)-\alpha^{2}R_{C}\left(1,1-\alpha% ^{2}\right),

-\infty<\alpha^{2}<1

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

… ►They are useful primarily when

\ifrac{(1-k)}{(1-\sin\phi)}

is either small or large compared with 1. … ►

19.12.6

R_{C}\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(% \sqrt{\frac{x}{y}}\right)\right),

x/y\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\pi$ : the ratio of the circumference of a circle to its diameter
Referenced by:: §19.12
Permalink:: http://dlmf.nist.gov/19.12.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19

►

19.12.7

R_{C}\left(x,y\right)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln% \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)\*(1+O\left(y^{2}/x^{2}\right)),

y/x\to 0

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\ln\NVar{z}$ : principal branch of logarithm function
Referenced by:: §19.12, §19.2(iv)
Permalink:: http://dlmf.nist.gov/19.12.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.12 and Ch.19