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11: 19.36 Methods of Computation

… ►For both

R_{D}

and

R_{J}

the factor

(r/4)^{-1/6}

in Carlson (1995, (2.18)) is changed to

(r/5)^{-1/8}

when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms: … ►Because of cancellations in (19.26.21) it is advisable to compute

R_{G}

from

R_{F}

and

R_{D}

by (19.21.10) or else to use §19.36(ii). … ►Accurate values of

F\left(\phi,k\right)-E\left(\phi,k\right)

for

k^{2}

near 0 can be obtained from

R_{D}

by (19.2.6) and (19.25.13). … ►

E\left(\phi,k\right)

can be evaluated by using (19.25.7), and

R_{D}

by using (19.21.10), but cancellations may become significant. … ►Near these points there will be loss of significant figures in the computation of

R_{J}

R_{D}

. …

12: 19.29 Reduction of General Elliptic Integrals

… ►For example, 3. …where the arguments of the

R_{D}

function are, in order,

(a-b)(u-c)

(b-c)(a-u)

(a-b)(b-c)

. … ►In the cubic case (

h=3

) the basic integrals are … ►If

h=3

, then the recurrence relation (Carlson (1999, (3.5))) has the special case … ►For example, because

t^{3}-a^{3}=(t-a)(t^{2}+at+a^{2})

, we find that when

0\leq a\leq y<x

…

13: 19.39 Software

… ►For research software see Bulirsch (1965b, function

\operatorname{el2}

), Bulirsch (1969b, function

\operatorname{el3}

), Jefferson (1961), and Neuman (1969a, functions

E\left(\phi,k\right)

and

\Pi\left(\phi,k^{2},k\right)

). …

14: 19.30 Lengths of Plane Curves

… ►

19.30.3

s/a=E\left(\phi,k\right)={R_{F}\left(c-1,c-k^{2},c\right)-\tfrac{1}{3}k^{2}R_{% D}\left(c-1,c-k^{2},c\right)},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, $k$ : real or complex modulus and $s$ : arclength
Referenced by:: §19.30(i)
Permalink:: http://dlmf.nist.gov/19.30.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(i), §19.30 and Ch.19

… ►

19.30.9

s=\tfrac{1}{2}I(\mathbf{e}_{1})=-\tfrac{1}{3}a^{2}b^{2}R_{D}\left(r,r+b^{2}+a^% {2},r+b^{2}\right)+y\sqrt{\frac{r+b^{2}+a^{2}}{r+b^{2}}},

r=b^{4}/y^{2}

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $I(\mathbf{m})$ : integral and $s$ : arclength
Permalink:: http://dlmf.nist.gov/19.30.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.30(ii), §19.30 and Ch.19

►For

s

in terms of

E\left(\phi,k\right)

F\left(\phi,k\right)

, and an algebraic term, see Byrd and Friedman (1971, p. 3). …

15: 19.26 Addition Theorems

… ►

19.26.7

R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+R_{D}\left(x+\mu,y+\mu,z+\mu% \right)=R_{D}\left(x,y,z\right)-\frac{3}{\sqrt{z(z+\lambda)(z+\mu)}},

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, $x$ : positive, $y$ : positive, $z$ : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.9

R_{J}\left(x+\lambda,y+\lambda,z+\lambda,p+\lambda\right)+R_{J}\left(x+\mu,y+% \mu,z+\mu,p+\mu\right)=R_{J}\left(x,y,z,p\right)-3R_{C}\left(\gamma-\delta,% \gamma\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\lambda$ : positive, $\delta$ , $x$ : positive, $y$ : positive, $z$ : positive, $\mu>0$ : parameter and $\gamma$
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.26.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(i), §19.26 and Ch.19

… ►

19.26.20

R_{D}\left(x,y,z\right)=2R_{D}\left(x+\lambda,y+\lambda,z+\lambda\right)+\frac% {3}{\sqrt{z}(z+\lambda)}.

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

… ►

19.26.22

R_{J}\left(x,y,z,p\right)=2R_{J}\left(x+\lambda,y+\lambda,z+\lambda,p+\lambda% \right)+3R_{C}\left(\alpha^{2},\beta^{2}\right),

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\lambda$ : positive, $\alpha$ , $\beta$ , $x$ : positive, $y$ : positive and $z$ : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §19.26(iii), §19.26 and Ch.19

…

16: 19.33 Triaxial Ellipsoids

… ►The surface area of an ellipsoid with semiaxes

a, b, c

, and volume

V=4\pi abc/3

is given by ►

19.33.1

S=3VR_{G}\left(a^{-2},b^{-2},c^{-2}\right),

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $V$ : volume and $S$ : area
Referenced by:: §19.33(i)
Permalink:: http://dlmf.nist.gov/19.33.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(i), §19.33 and Ch.19

… ►Let a homogeneous magnetic ellipsoid with semiaxes

a, b, c

, volume

V=4\pi abc/3

, and susceptibility

\chi

be placed in a previously uniform magnetic field

H

parallel to the principal axis with semiaxis

c

. … ►

19.33.7

L_{c}=2\pi abc\int_{0}^{\infty}\frac{\,\mathrm{d}\lambda}{\sqrt{(a^{2}+\lambda% )(b^{2}+\lambda)(c^{2}+\lambda)^{3}}}=VR_{D}\left(a^{2},b^{2},c^{2}\right).

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $L_{a}$ , $L_{b}$ : factor and $V$ : volume
Referenced by:: §19.15, §19.33(iii)
Permalink:: http://dlmf.nist.gov/19.33.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §19.33(iii), §19.33 and Ch.19

…

17: 19.1 Special Notation

… ►

R_{D}\left(x,y,z\right)

… ►

\operatorname{el3}\left(x,k_{c},p\right),

…

18: 19.24 Inequalities

… ►Inequalities for

R_{D}\left(0,y,z\right)

are included as the case

p=z

. … ►Other inequalities can be obtained by applying Carlson (1966, Theorems 2 and 3) to (19.16.20)–(19.16.23). … ►Inequalities for

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

in Carlson (1966, Theorems 2 and 3) can be applied to (19.16.14)–(19.16.17). … ►

19.24.12

\tfrac{1}{3}(\sqrt{x}+\sqrt{y}+\sqrt{z})\leq R_{G}\left(x,y,z\right)\leq\min% \left(\sqrt{\frac{x+y+z}{3}},\frac{x^{2}+y^{2}+z^{2}}{3\sqrt{xyz}}\right).

ⓘ

Symbols:: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind
Permalink:: http://dlmf.nist.gov/19.24.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

►Inequalities for

R_{C}\left(x,y\right)

and

R_{D}\left(x,y,z\right)

are included as special cases (see (19.16.6) and (19.16.5)). …

19: 19.23 Integral Representations

… ►

19.23.3

R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)% ^{-3/2}{\sin}^{2}\theta\,\mathrm{d}\theta.

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

… ►In (19.23.8)

n=2

, and in (19.23.9)

n=3

. … ►With

l_{1},l_{2},l_{3}

denoting any permutation of

\sin\theta\cos\phi

\sin\theta\sin\phi

\cos\theta

, ►

19.23.9

R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4\Gamma\left(b_{1}+b_{2}+b_{3}% \right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\Gamma\left(b_{3}% \right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left(\sum_{j=1}^{3}z_{j}l_{j}% ^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin\theta\,\mathrm{d}\theta% \,\mathrm{d}\phi,

b_{j}>0

\Re z_{j}>0

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $l$ : nonnegative integer and $\phi$ : real or complex argument
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19

…

20: 19.19 Taylor and Related Series

… ►

19.19.6

R_{J}\left(x,y,z,p\right)=R_{-\frac{3}{2}}\left(\tfrac{1}{2},\tfrac{1}{2},% \tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z,p,p\right)

ⓘ

Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.19
Permalink:: http://dlmf.nist.gov/19.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §19.19 and Ch.19

… ►For

R_{J}

and

R_{D}

T_{N}

has at most one term if

N\leq 3

, and two terms if

N=4

or 5. …