Bulirsch elliptic integrals

SearchAdvancedHelp

(0.001 seconds)

8 matching pages

1: 19.2 Definitions

… ►

§19.2(iii) Bulirsch’s Integrals

… ►

19.2.11_5 $$\mathrm{el1}(x,k_c)={\int }_0^{\arctan x}\frac{1}{\sqrt{{\cos }^2\theta +k_c^2{\sin }^2\theta }}d\theta ,$$

ⓘ

Defines:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind

Symbols:

$\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\arctan z$: arctangent function, $\sin z$: sine function, $k_c$: real or complex variable not equal to zero and $x$: real or complex variable

Referenced by:

§19.2(iii), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/19.2.E11_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added and made the definition for $\mathrm{el1}(x,k_c)$ (was previously (19.2.15)).

Suggested 2019-09-26 by Jan Mangaldan

See also:

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

… ►

$K\left(k\right)=\mathrm{cel}(k_c,1,1,1),$

►

$E\left(k\right)=\mathrm{cel}(k_c,1,1,k_c^2),$

►

$D\left(k\right)=\mathrm{cel}(k_c,1,0,1),$

…

2: 19.1 Special Notation

… ► ►►

$l,m,n$	nonnegative integers.
…

… ► $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

3: 19.25 Relations to Other Functions

… ►

§19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ►

19.25.19 $$\mathrm{cel}(k_c,p,a,b)=a{R}_F(0,k_c^2,1)+\frac{1}{3}(b-pa)R_J(0,k_c^2,1,p),$$

ⓘ

Symbols:

$\mathrm{cel}(k_c,p,a,b)$: Bulirsch’s complete elliptic integral, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.25.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

►

19.25.20 $$\mathrm{el1}(x,k_c)=R_F(r,r+k_c^2,r+1),$$

ⓘ

Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

►

19.25.21 $$\mathrm{el2}(x,k_c,a,b)=a\mathrm{el1}(x,k_c)+\frac{1}{3}(b-a)R_D(r,r+k_c^2,r+1),$$

ⓘ

Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el2}(x,k_c,a,b)$: Bulirsch’s incomplete elliptic integral of the second kind, $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

►

19.25.22 $$\mathrm{el3}(x,k_c,p)=\mathrm{el1}(x,k_c)+\frac{1}{3}(1-p)R_J(r,r+k_c^2,r+1,r+p).$$

ⓘ

Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $\mathrm{el3}(x,k_c,p)$: Bulirsch’s incomplete elliptic integral of the third kind, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

…

4: 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 $\mathrm{\Pi }(\varphi ,k^2,k)$). …

5: 19.36 Methods of Computation

… ►Computation of Legendre’s integrals of all three kinds by quadratic transformation is described by Cazenave (1969, pp. 128–159, 208–230). …

6: Bibliography B

… ►

An extension of the Bartky-transformation to incomplete elliptic integrals of the third kind. Numer. Math. 13 (3), pp. 266–284.

ⓘ

External Links:

ISSN 0029-599X, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.36(ii).

Encodings:

BibTeX

►

Numerical calculation of elliptic integrals and elliptic functions. III. Numer. Math. 13 (4), pp. 305–315.

ⓘ

Notes:

Algol 60 programs for the incomplete elliptic integral of the third kind, and for a general complete elliptic integral, are included.

External Links:

ISSN 0029-599X, Document, MathReview, ZentralBlatt

Referenced by:

§19.2(iii), §19.36(ii), §19.36(ii), §19.39(ii), §19.39(iii), R. Bulirsch (1965a).

Encodings:

BibTeX

… ►

Numerical calculation of elliptic integrals and elliptic functions. II. Numer. Math. 7 (4), pp. 353–354.

ⓘ

Notes:

An improved Algol 60 program for the complete elliptic integral of the third kind is included, but superseded by a program in Bulirsch (1969b).

External Links:

ISSN 0029-599X, Document, MathReview (M. P. S. Madan), ZentralBlatt

Referenced by:

§19.1.

Encodings:

BibTeX

►

Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7 (1), pp. 78–90.

ⓘ

Notes:

Algol 60 programs are included for real and complex elliptic integrals, and for Jacobian elliptic functions of real argument.

External Links:

ISSN 0029-599X, Document, MathReview (M. Feliciano), ZentralBlatt

Referenced by:

§19.1, §19.2(iii), §19.36(ii), §19.39(iii), §22.22(ii).

Encodings:

BibTeX

…

7: Bibliography R

… ►

Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

Encodings:

BibTeX

…

8: Bibliography M

… ►

An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.

ⓘ

External Links:

ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

…