19 Elliptic Integrals Legendre’s Integrals 19.11 Addition Theorems 19.13 Integrals of Elliptic Integrals

§19.12 Asymptotic Approximations

ⓘ

Keywords:: Legendre’s elliptic integrals, $R_{C}$ -function, asymptotic approximations
Notes:: For (19.12.1) and (19.12.2) see Cayley (1895, p. 54) and Cazenave (1969, pp. 165–169). For (19.12.4) and (19.12.5) use (19.25.2), (19.27.13), and (19.6.5). For (19.12.6) and (19.12.7) see Carlson and Gustafson (1994, (22), (24)).
Referenced by:: §19.6(i), §22.3(ii), §22.5(ii), Erratum (V1.0.3) for References, Erratum (V1.0.5) for References, Erratum (V1.0.5) for Clarifications
Permalink:: http://dlmf.nist.gov/19.12
Addition and Clarification (effective with 1.0.5):: The reference to Karp and Sitnik (2007) has been added in the middle of this section. Also, the definition of $d(0)$ was further clarified (made explicit) after (19.12.3).
Clarification (effective with 1.0.3):: The definition of $d(0)$ was clarified after (19.12.3).
See also:: Annotations for Ch.19

With $\psi\left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series:

19.12.1		$K\left(k\right)=\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{m}}{\left% (\tfrac{1}{2}\right)_{m}}}{m!\;m!}{k^{\prime}}^{2m}\left(\ln\left(\frac{1}{k^{% \prime}}\right)+d(m)\right),$
$0<\|k^{\prime}\|<1$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E1 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19

19.12.2		$E\left(k\right)=1+\frac{1}{2}\sum_{m=0}^{\infty}\frac{{\left(\tfrac{1}{2}% \right)_{m}}{\left(\tfrac{3}{2}\right)_{m}}}{{\left(2\right)_{m}}m!}{k^{\prime% }}^{2m+2}\*\left(\ln\left(\frac{1}{k^{\prime}}\right)+d(m)-\frac{1}{(2m+1)(2m+% 2)}\right),$
$\|k^{\prime}\|<1$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $m$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $d(m)$ : function Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E2 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19

where

19.12.3		$\displaystyle d(m)$	$\displaystyle=\psi\left(1+m\right)-\psi\left(\tfrac{1}{2}+m\right),$
		$\displaystyle d(m+1)$	$\displaystyle=d(m)-\frac{2}{(2m+1)(2m+2)}$ ,
	$m=0,1,\dots$ ,
ⓘ Symbols: $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $m$ : nonnegative integer and $d(m)$ : function Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E3 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.12 and Ch.19

with $d(0)=2\ln 2$ .

For the asymptotic behavior of $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ as $\phi\to\tfrac{1}{2}\pi-$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007).

As $k^{2}\to 1-$

19.12.4		$(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\frac{4}{k^{\prime}}\right% )\left(1+O\left({k^{\prime}}^{2}\right)\right)-\alpha^{2}R_{C}\left(1,1-\alpha% ^{2}\right),$
$-\infty<\alpha^{2}<1$ ,
ⓘ 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, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E4 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19

19.12.5		$(1-\alpha^{2})\Pi\left(\alpha^{2},k\right)=\left(\ln\left(\frac{4}{k^{\prime}}% \right)-R_{C}\left(1,1-\alpha^{-2}\right)\right)\*\left(1+O\left({k^{\prime}}^% {2}\right)\right),$
$1<\alpha^{2}<\infty$ .
ⓘ 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, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E5 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19

Asymptotic approximations for $\Pi\left(\phi,\alpha^{2},k\right)$ , with different variables, are given in Karp et al. (2007). They are useful primarily when $\ifrac{(1-k)}{(1-\sin\phi)}$ is either small or large compared with 1.

If $x\geq 0$ and $y>0$ , then

19.12.6		$R_{C}\left(x,y\right)=\frac{\pi}{2\sqrt{y}}-\frac{\sqrt{x}}{y}\left(1+O\left(% \sqrt{\frac{x}{y}}\right)\right),$
$x/y\to 0$ ,
ⓘ 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 and $\pi$ : the ratio of the circumference of a circle to its diameter Referenced by: §19.12 Permalink: http://dlmf.nist.gov/19.12.E6 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19

19.12.7		$R_{C}\left(x,y\right)=\frac{1}{2\sqrt{x}}\left(\left(1+\frac{y}{2x}\right)\ln% \left(\frac{4x}{y}\right)-\frac{y}{2x}\right)\*(1+O\left(y^{2}/x^{2}\right)),$
$y/x\to 0$ .
ⓘ 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 and $\ln\NVar{z}$ : principal branch of logarithm function Referenced by: §19.12, §19.2(iv) Permalink: http://dlmf.nist.gov/19.12.E7 Encodings: TeX, pMML, png See also: Annotations for §19.12 and Ch.19

19.11 Addition Theorems 19.13 Integrals of Elliptic Integrals

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