19 Elliptic Integrals Symmetric Integrals 19.26 Addition Theorems 19.28 Integrals of Elliptic Integrals

§19.27 Asymptotic Approximations and Expansions

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Keywords:: asymptotic approximations and expansions, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.27
See also:: Annotations for Ch.19

§19.27(i) Notation

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Referenced by:: §19.27(vi)
Permalink:: http://dlmf.nist.gov/19.27.i
See also:: Annotations for §19.27 and Ch.19

Throughout this section

19.27.1	$\displaystyle a$	$\displaystyle=\tfrac{1}{2}(x+y),$
	$\displaystyle b$	$\displaystyle=\tfrac{1}{2}(y+z),$
	$\displaystyle c$	$\displaystyle=\tfrac{1}{3}(x+y+z),$
	$\displaystyle f$	$\displaystyle=(xyz)^{1/3},$
	$\displaystyle g$	$\displaystyle=(xy)^{1/2},$
	$\displaystyle h$	$\displaystyle=(yz)^{1/2}.$
ⓘ Symbols: $a$ , $b$ , $c$ , $f$ , $g$ and $h$ Permalink: http://dlmf.nist.gov/19.27.E1 Encodings: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png See also: Annotations for §19.27(i), §19.27 and Ch.19

§19.27(ii) $R_{F}\left(x,y,z\right)$

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Notes:: Adapted from Carlson and Gustafson (1994, (26), (32)). With regard to (19.27.2) see also Carlson and Gustafson (1985, (1.15), (1.16), and (3.46)).
Permalink:: http://dlmf.nist.gov/19.27.ii
See also:: Annotations for §19.27 and Ch.19

Assume $x$ , $y$ , and $z$ are real and nonnegative and at most one of them is 0. Then

19.27.2		$R_{F}\left(x,y,z\right)=\frac{1}{2\sqrt{z}}\left(\ln\frac{8z}{a+g}\right)\left% (1+O\left(\frac{a}{z}\right)\right),$
$a/z\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ and $g$ Referenced by: §19.20(iii), §19.27(ii) Permalink: http://dlmf.nist.gov/19.27.E2 Encodings: TeX, pMML, png See also: Annotations for §19.27(ii), §19.27 and Ch.19

19.27.3		$R_{F}\left(x,y,z\right)=R_{F}\left(0,y,z\right)-\frac{1}{\sqrt{h}}\left(\sqrt{% \frac{x}{h}}+O\left(\frac{x}{h}\right)\right),$
$x/h\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $h$ Permalink: http://dlmf.nist.gov/19.27.E3 Encodings: TeX, pMML, png See also: Annotations for §19.27(ii), §19.27 and Ch.19

§19.27(iii) $R_{G}\left(x,y,z\right)$

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Notes:: Adapted from Carlson and Gustafson (1994, (51), (54), (52)).
Permalink:: http://dlmf.nist.gov/19.27.iii
See also:: Annotations for §19.27 and Ch.19

Assume $x$ , $y$ , and $z$ are real and nonnegative and at most one of them is 0. Then

19.27.4		$R_{G}\left(x,y,z\right)=\frac{\sqrt{z}}{2}\left(1+O\left(\frac{a}{z}\ln\frac{z% }{a}\right)\right),$
$a/z\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\ln\NVar{z}$ : principal branch of logarithm function and $a$ Permalink: http://dlmf.nist.gov/19.27.E4 Encodings: TeX, pMML, png See also: Annotations for §19.27(iii), §19.27 and Ch.19

19.27.5		$R_{G}\left(x,y,z\right)=R_{G}\left(0,y,z\right)+\sqrt{x}O\left(\sqrt{x/h}% \right),$
$x/h\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $h$ Permalink: http://dlmf.nist.gov/19.27.E5 Encodings: TeX, pMML, png See also: Annotations for §19.27(iii), §19.27 and Ch.19

19.27.6		$R_{G}\left(0,y,z\right)={\frac{\sqrt{z}}{2}+\frac{y}{8\sqrt{z}}\left(\ln\left(% \frac{16z}{y}\right)-1\right)\*\left(1+O\left(\frac{y}{z}\right)\right)},$
$y/z\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $\ln\NVar{z}$ : principal branch of logarithm function Permalink: http://dlmf.nist.gov/19.27.E6 Encodings: TeX, pMML, png See also: Annotations for §19.27(iii), §19.27 and Ch.19

§19.27(iv) $R_{D}\left(x,y,z\right)$

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Notes:: Adapted from Carlson and Gustafson (1994, (34), (37)–(39)).
Permalink:: http://dlmf.nist.gov/19.27.iv
See also:: Annotations for §19.27 and Ch.19

Assume $x$ and $y$ are real and nonnegative, at most one of them is 0, and $z>0$ . Then

19.27.7	$\displaystyle R_{D}\left(x,y,z\right)$	$\displaystyle=\frac{3}{2z^{3/2}}\left(\ln\left(\frac{8z}{a+g}\right)-2\right)% \left(1+O\left(\frac{a}{z}\right)\right),$
$a/z\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ and $g$ Referenced by: §19.20(iv) Permalink: http://dlmf.nist.gov/19.27.E7 Encodings: TeX, pMML, png See also: Annotations for §19.27(iv), §19.27 and Ch.19
19.27.8	$\displaystyle R_{D}\left(x,y,z\right)$	$\displaystyle=\frac{3}{\sqrt{xyz}}-\frac{6}{xy}R_{G}\left(x,y,0\right)\left(1+% O\left(\frac{z}{g}\right)\right),$
$z/g\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $g$ Referenced by: §19.20(iv), §19.29(iii) Permalink: http://dlmf.nist.gov/19.27.E8 Encodings: TeX, pMML, png See also: Annotations for §19.27(iv), §19.27 and Ch.19
19.27.9	$\displaystyle R_{D}\left(x,y,z\right)$	$\displaystyle=\frac{3}{\sqrt{xz}(\sqrt{y}+\sqrt{z})}\left(1+O\left(\frac{b}{x}% \ln\frac{x}{b}\right)\right),$
$b/x\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\ln\NVar{z}$ : principal branch of logarithm function and $b$ Permalink: http://dlmf.nist.gov/19.27.E9 Encodings: TeX, pMML, png See also: Annotations for §19.27(iv), §19.27 and Ch.19
19.27.10	$\displaystyle R_{D}\left(x,y,z\right)$	$\displaystyle=R_{D}\left(0,y,z\right)-\frac{3\sqrt{x}}{hz}\left(1+O\left(\sqrt% {\frac{x}{h}}\right)\right),$
$x/h\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $h$ Permalink: http://dlmf.nist.gov/19.27.E10 Encodings: TeX, pMML, png See also: Annotations for §19.27(iv), §19.27 and Ch.19

§19.27(v) $R_{J}\left(x,y,z,p\right)$

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Notes:: Adapted from Carlson and Gustafson (1994, (40), (42), (44), (47)–(49)).
Referenced by:: §19.27(vi)
Permalink:: http://dlmf.nist.gov/19.27.v
See also:: Annotations for §19.27 and Ch.19

Assume $x$ , $y$ , and $z$ are real and nonnegative, at most one of them is 0, and $p>0$ . Then

19.27.11	$\displaystyle R_{J}\left(x,y,z,p\right)$	$\displaystyle={\frac{3}{p}R_{F}\left(x,y,z\right)-\frac{3\pi}{2p^{3/2}}\left(1% +O\left(\sqrt{\frac{c}{p}}\right)\right)},$
$c/p\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\pi$ : the ratio of the circumference of a circle to its diameter and $c$ Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.27.E11 Encodings: TeX, pMML, png See also: Annotations for §19.27(v), §19.27 and Ch.19
19.27.12	$\displaystyle R_{J}\left(x,y,z,p\right)$	$\displaystyle={\frac{3}{2\sqrt{xyz}}\left(\ln\left(\frac{4f}{p}\right)-2\right% )\left(1+O\left(\frac{p}{f}\right)\right)},$
$p/f\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\ln\NVar{z}$ : principal branch of logarithm function and $f$ Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.27.E12 Encodings: TeX, pMML, png See also: Annotations for §19.27(v), §19.27 and Ch.19
19.27.13	$\displaystyle R_{J}\left(x,y,z,p\right)$	$\displaystyle=\frac{3}{2\sqrt{z}p}\left(\ln\left(\frac{8z}{a+g}\right)-2R_{C}% \left(1,\frac{p}{z}\right)+O\left(\left(\frac{a}{z}+\frac{a}{p}\right)\ln\frac% {p}{a}\right)\right),$
$\max(x,y)/\min(z,p)\to 0$ .
ⓘ 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, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $a$ and $g$ Referenced by: §19.12, §19.20(iii) Permalink: http://dlmf.nist.gov/19.27.E13 Encodings: TeX, pMML, png See also: Annotations for §19.27(v), §19.27 and Ch.19
19.27.14	$\displaystyle R_{J}\left(x,y,z,p\right)$	$\displaystyle=\frac{3}{\sqrt{yz}}R_{C}\left(x,p\right)-\frac{6}{yz}R_{G}\left(% 0,y,z\right)+O\left(\frac{\sqrt{x+2p}}{yz}\right),$
$\max(x,p)/\min(y,z)\to 0$ .
ⓘ 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, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind Referenced by: §19.20(iii) Permalink: http://dlmf.nist.gov/19.27.E14 Encodings: TeX, pMML, png See also: Annotations for §19.27(v), §19.27 and Ch.19
19.27.15	$\displaystyle R_{J}\left(x,y,z,p\right)$	$\displaystyle=R_{J}\left(0,y,z,p\right)-\frac{3\sqrt{x}}{hp}\left(1+O\left(% \left(\frac{b}{h}+\frac{h}{p}\right)\sqrt{\frac{x}{h}}\right)\right),$
$x/\min(y,z,p)\to 0$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $b$ and $h$ Permalink: http://dlmf.nist.gov/19.27.E15 Encodings: TeX, pMML, png See also: Annotations for §19.27(v), §19.27 and Ch.19
19.27.16	$\displaystyle R_{J}\left(x,y,z,p\right)$	$\displaystyle=(3/\sqrt{x})R_{C}\left((h+p)^{2},2(b+h)p\right)+O\left(\frac{1}{% x^{3/2}}\ln\frac{x}{b+h}\right),$
$\max(y,z,p)/x\to 0$ .
ⓘ 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, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $b$ and $h$ Permalink: http://dlmf.nist.gov/19.27.E16 Encodings: TeX, pMML, png See also: Annotations for §19.27(v), §19.27 and Ch.19

§19.27(vi) Asymptotic Expansions

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Keywords:: asymptotic approximations and expansions, symmetric elliptic integrals
Referenced by:: §2.6(ii)
Permalink:: http://dlmf.nist.gov/19.27.vi
See also:: Annotations for §19.27 and Ch.19

The approximations in §§19.27(i)–19.27(v) are furnished with upper and lower bounds by Carlson and Gustafson (1994), sometimes with two or three approximations of differing accuracies. Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)). These series converge but not fast enough, given the complicated nature of their terms, to be very useful in practice.

A similar (but more general) situation prevails for $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$ when some of the variables $z_{1},\dots,z_{n}$ are smaller in magnitude than the rest; see Carlson (1985, (4.16)–(4.19) and (2.26)–(2.29)).

19.26 Addition Theorems 19.28 Integrals of Elliptic Integrals

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§19.27 Asymptotic Approximations and Expansions

Contents

§19.27(i) Notation

§19.27(ii) $R_{F}\left(x,y,z\right)$

§19.27(iii) $R_{G}\left(x,y,z\right)$

§19.27(iv) $R_{D}\left(x,y,z\right)$

§19.27(v) $R_{J}\left(x,y,z,p\right)$

§19.27(vi) Asymptotic Expansions

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§19.27 Asymptotic Approximations and Expansions

Contents

§19.27(i) Notation

§19.27(ii) RF⁡(x,y,z)

§19.27(iii) RG⁡(x,y,z)

§19.27(iv) RD⁡(x,y,z)

§19.27(v) RJ⁡(x,y,z,p)

§19.27(vi) Asymptotic Expansions

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§19.27(ii) $R_{F}\left(x,y,z\right)$

§19.27(iii) $R_{G}\left(x,y,z\right)$

§19.27(iv) $R_{D}\left(x,y,z\right)$

§19.27(v) $R_{J}\left(x,y,z,p\right)$