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11: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►See Srinivasa Rao and Rajeswari (1993, pp. 102–103) and references given there. …

12: Bibliography O

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Error Analysis of Complex Arithmetic. In Computational Aspects of Complex Analysis (Braunlage, 1982), H. Werner, L. Wuytack, E. Ng, and H. J. Bünger (Eds.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 102, pp. 279–292.

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External Links:

MathReview (G. Blanch), ZentralBlatt

Referenced by:

§3.1(v).

Encodings:

BibTeX

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13: Bibliography T

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Simplified calculation of $\mathrm{Ei}(x)$ for positive arguments, and a short table of $\mathrm{Shi}(x)$ . Math. Comp. 22 (102), pp. 448–449.

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External Links:

FullText, Document, ZentralBlatt

Referenced by:

§6.18(i).

Encodings:

BibTeX

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14: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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External Links:

ISSN 0024-1318, Document, MathReview (R. A. Askey)

Referenced by:

§18.7(iii).

Encodings:

BibTeX

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Orthogonalization Procedures and the Localization of Wannier Functions. Phys. Rev. 105, pp. 102–103.

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External Links:

Document, Link, ZentralBlatt

Referenced by:

Profile Bille C. Carlson.

Encodings:

BibTeX

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Chebyshev approximations for the Fresnel integrals. Math. Comp. 22 (102), pp. 450–453.

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Notes:

with Microfiche Supplement

External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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15: Bibliography H

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Some properties and applications of the repeated integrals of the error function. Proc. Manchester Lit. Philos. Soc. 80, pp. 85–102.

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External Links:

ZentralBlatt

Referenced by:

§7.18(iii), §7.18(iv), §7.18(v), §7.18(vi).

Encodings:

BibTeX

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16: Bibliography W

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Recursion formulae for hypergeometric functions. Math. Comp. 22 (102), pp. 363–373.

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External Links:

FullText, Document, MathReview (S. D. Bajpai), ZentralBlatt

Referenced by:

§16.3(ii).

Encodings:

BibTeX

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17: 19.39 Software

… ►For research software see Bulirsch (1965b, function $\mathrm{el2}$), Bulirsch (1969b, function $\mathrm{el3}$), Jefferson (1961), and Neuman (1969a, functions $E(\varphi ,k)$ and $\mathrm{\Pi }(\varphi ,k^2,k)$). …

18: 19.1 Special Notation

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$\mathrm{el2}(x,k_c,a,b),$

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19: 19.2 Definitions

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19.2.12 $$\mathrm{el2}(x,k_c,a,b)={\int }_0^{\arctan x}\frac{a+b{\tan }^2\theta }{\sqrt{(1+{\tan }^2\theta )(1+k_c^2{\tan }^2\theta )}}d\theta .$$

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Defines:

$\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\arctan z$: arctangent function, $\tan z$: tangent function, $a$: real parameter, $b$: real parameter, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable

Permalink:

http://dlmf.nist.gov/19.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(iii), §19.2 and Ch.19

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$$F(\varphi ,k)=\mathrm{el1}(x,k_c)=\mathrm{el2}(x,k_c,1,1),$$

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$$E(\varphi ,k)=\mathrm{el2}(x,k_c,1,k_c^2),$$

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$$D(\varphi ,k)=\mathrm{el2}(x,k_c,0,1).$$

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20: 19.36 Methods of Computation

… ►The function $\mathrm{el2}(x,k_c,a,b)$ is computed by descending Landen transformations if $x$ is real, or by descending Gauss transformations if $x$ is complex (Bulirsch (1965b)). …