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1: 19.1 Special Notation

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$\mathrm{el1}(x,k_c),$

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

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$\mathrm{el3}(x,k_c,p),$

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

3: 19.2 Definitions

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

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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

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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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4: 19.25 Relations to Other Functions

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

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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

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

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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

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

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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

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5: 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)). … ►Bulirsch (1969a, b) extend Bartky’s transformation to $\mathrm{el3}(x,k_c,p)$ by expressing it in terms of the first incomplete integral, a complete integral of the third kind, and a more complicated integral to which Bartky’s method can be applied. …

6: Bibliography M

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A Course in Real Analysis. Academic Press Inc., San Diego, CA.

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External Links:

ISBN 0-12-742830-5, MathReview (Sherif T. El-Helaly), ZentralBlatt

Referenced by:

§1.4(v), §18.2(i).

Encodings:

BibTeX

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