19 Elliptic Integrals Legendre’s Integrals 19.8 Quadratic Transformations 19.10 Relations to Other Functions

§19.9 Inequalities

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Referenced by:: Ch.19
Permalink:: http://dlmf.nist.gov/19.9
See also:: Annotations for Ch.19

§19.9(i) Complete Integrals
§19.9(ii) Incomplete Integrals

§19.9(i) Complete Integrals

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Keywords:: Legendre’s elliptic integrals, complete integrals, inequalities
Notes:: For (19.9.1) see Erdélyi et al. (1953b, §13.8(9), (11)), (19.9.13), (19.6.12), and (19.6.15). For (19.9.2) and (19.9.3) see Qiu and Vamanamurthy (1996). For (19.9.4) see Barnard et al. (2000, (6)); the first inequality was given earlier by Qiu and Shen (1997, Theorem 2). For (19.9.5) see Lehto and Virtanen (1973, p. 62). For (19.9.6) and (19.9.7) see (19.25.1) and (19.16.21) and then apply Carlson (1966, (2.15)), in which $H<H^{\prime}$ for $0<k\leq 1$ in both cases. In (19.9.7) the upper bound $4/\pi$ , which is the smaller of the two when $k^{2}\geq 0.855\dots$ , is given by Anderson and Vamanamurthy (1985). For (19.9.8) see (19.25.1), Neuman (2003, (4.2)), and (19.24.9). For (19.9.9) see (19.30.5).
Referenced by:: §19.24(i), §19.30(i)
Permalink:: http://dlmf.nist.gov/19.9.i
See also:: Annotations for §19.9 and Ch.19

Throughout this subsection $0<k<1$ , except in (19.9.4).

19.9.1		$\displaystyle\ln 4$	$\displaystyle\leq K\left(k\right)+\ln k^{\prime}\leq\pi/2,$
		$\displaystyle 1$	$\displaystyle\leq E\left(k\right)\leq\pi/2.$
		$\displaystyle 1$	$\displaystyle\leq(2/\pi)\sqrt{1-\alpha^{2}}\Pi\left(\alpha^{2},k\right)\leq 1/% k^{\prime},$
	$\alpha^{2}<1$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : 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: §19.24(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E1 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.9(i), §19.9 and Ch.19

19.9.2		$1+\frac{{k^{\prime}}^{2}}{8}<\frac{K\left(k\right)}{\ln\left(4/k^{\prime}% \right)}<1+\frac{{k^{\prime}}^{2}}{4},$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.9(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E2 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

19.9.3		$9+\frac{k^{2}{k^{\prime}}^{2}}{8}<\frac{(8+k^{2})K\left(k\right)}{\ln\left(4/k% ^{\prime}\right)}<9.096.$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.9(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E3 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

The left-hand inequalities in (19.9.2) and (19.9.3) are equivalent, but the right-hand inequality of (19.9.3) is sharper than that of (19.9.2) when $0<k^{2}\leq 0.922$ .

19.9.4		$\left(\frac{1+{k^{\prime}}^{3/2}}{2}\right)^{2/3}\leq\frac{2}{\pi}E\left(k% \right)\leq\left(\frac{1+{k^{\prime}}^{2}}{2}\right)^{1/2}$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.24(i), §19.9(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E4 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

for $0\leq k\leq 1$ . The lower bound in (19.9.4) is sharper than $2/\pi$ when $0\leq k^{2}\leq 0.9960$ .

19.9.5		$\ln\frac{(1+\sqrt{k^{\prime}})^{2}}{k}<\frac{\pi{K^{\prime}}\left(k\right)}{2K% \left(k\right)}<\ln\frac{2(1+k^{\prime})}{k}.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.9(i), §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E5 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

For a sharper, but more complicated, version of (19.9.5) see Anderson et al. (1990).

Other inequalities are:

19.9.6		$(1-\tfrac{3}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(K\left(k\right)-E\left(k% \right))<(k^{\prime})^{-3/4},$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E6 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

19.9.7		$(1-\tfrac{1}{4}k^{2})^{-1/2}<\frac{4}{\pi k^{2}}(E\left(k\right)-{k^{\prime}}^% {2}K\left(k\right))<\min((k^{\prime})^{-1/4},4/\pi),$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E7 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

19.9.8		$k^{\prime}<\frac{E\left(k\right)}{K\left(k\right)}<\left(\frac{1+k^{\prime}}{2% }\right)^{2}.$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E8 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

Further inequalities for $K\left(k\right)$ and $E\left(k\right)$ can be found in Alzer and Qiu (2004), Anderson et al. (1992a, b, 1997), and Qiu and Vamanamurthy (1996).

The perimeter $L(a,b)$ of an ellipse with semiaxes $a, b$ is given by

19.9.9		$L(a,b)=4aE\left(k\right),$
$k^{2}=1-(b^{2}/a^{2})$ , $a>b$ .
ⓘ Symbols: $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus and $L(a,b)$ : perimeter Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E9 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

Almkvist and Berndt (1988) list thirteen approximations to $L(a,b)$ that have been proposed by various authors. The earliest is due to Kepler and the most accurate to Ramanujan. Ramanujan’s approximation and its leading error term yield the following approximation to $L(a,b)/(\pi(a+b))$ :

19.9.10		$1+\frac{3\lambda^{2}}{10+\sqrt{4-3\lambda^{2}}}+\frac{3\lambda^{10}}{2^{17}},$
$\lambda=\dfrac{a-b}{a+b}$ .
ⓘ Permalink: http://dlmf.nist.gov/19.9.E10 Encodings: TeX, pMML, png See also: Annotations for §19.9(i), §19.9 and Ch.19

Even for the extremely eccentric ellipse with $a=99$ and $b=1$ , this is correct within 0.023%. Barnard et al. (2000) shows that nine of the thirteen approximations, including Ramanujan’s, are from below and four are from above. See also Barnard et al. (2001).

§19.9(ii) Incomplete Integrals

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Keywords:: Legendre’s elliptic integrals, incomplete integrals, inequalities
Notes:: For (19.9.14) see (19.24.10) and (19.25.5). For (19.9.15) and (19.9.16) see Carlson and Gustafson (1985, (1.2), (1.22)).
Permalink:: http://dlmf.nist.gov/19.9.ii
See also:: Annotations for §19.9 and Ch.19

Throughout this subsection we assume that $0<k<1$ , $0\leq\phi\leq\pi/2$ , and $\Delta=\sqrt{1-k^{2}{\sin}^{2}\phi}>0$ .

Simple inequalities for incomplete integrals follow directly from the defining integrals (§19.2(ii)) together with (19.6.12):

19.9.11		$\phi\leq F\left(\phi,k\right)\leq\min(\phi/\Delta,{\operatorname{gd}^{-1}}% \left(\phi\right)),$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, ${\operatorname{gd}^{-1}}\left(\NVar{x}\right)$ : inverse Gudermannian function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$ Permalink: http://dlmf.nist.gov/19.9.E11 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

where ${\operatorname{gd}^{-1}}\left(\phi\right)$ is given by (4.23.41) and (4.23.42). Also,

19.9.12		$\max(\sin\phi,\phi\Delta)\leq E\left(\phi,k\right)\leq\phi,$
ⓘ Symbols: $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$ Permalink: http://dlmf.nist.gov/19.9.E12 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

19.9.13		$\Pi\left(\phi,\alpha^{2},0\right)\leq\Pi\left(\phi,\alpha^{2},k\right)\leq\min% (\Pi\left(\phi,\alpha^{2},0\right)/\Delta,\Pi\left(\phi,\alpha^{2},1\right)).$
ⓘ Symbols: $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $\alpha^{2}$ : real or complex parameter and $\Delta$ Referenced by: §19.9(i) Permalink: http://dlmf.nist.gov/19.9.E13 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

Sharper inequalities for $F\left(\phi,k\right)$ are:

19.9.14		$\frac{3}{1+\Delta+\cos\phi}<\frac{F\left(\phi,k\right)}{\sin\phi}<\frac{1}{(% \Delta\cos\phi)^{1/3}},$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$ Referenced by: §19.9(ii), §19.9(ii) Permalink: http://dlmf.nist.gov/19.9.E14 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

19.9.15		$1<F\left(\phi,k\right)\bigg{/}\left((\sin\phi)\ln\left(\frac{4}{\Delta+\cos% \phi}\right)\right)<\frac{4}{2+(1+k^{2}){\sin}^{2}\phi}.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$ Referenced by: §19.9(ii), §19.9(ii) Permalink: http://dlmf.nist.gov/19.9.E15 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

19.9.16		$F\left(\phi,k\right)=\frac{2}{\pi}K\left(k^{\prime}\right)\ln\left(\frac{4}{% \Delta+\cos\phi}\right)-\theta\Delta^{2},$
$(\sin\phi)/8<\theta<(\ln 2)/(k^{2}\sin\phi)$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\cos\NVar{z}$ : cosine function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\Delta$ Referenced by: §19.9(ii) Permalink: http://dlmf.nist.gov/19.9.E16 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

(19.9.15) is useful when $k^{2}$ and ${\sin}^{2}\phi$ are both close to $1$ , since the bounds are then nearly equal; otherwise (19.9.14) is preferable.

Inequalities for both $F\left(\phi,k\right)$ and $E\left(\phi,k\right)$ involving inverse circular or inverse hyperbolic functions are given in Carlson (1961b, §4). For example,

19.9.17		$L\leq F\left(\phi,k\right)\leq\sqrt{UL}\leq\tfrac{1}{2}(U+L)\leq U,$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $L$ : lower limit and $U$ : upper limit Permalink: http://dlmf.nist.gov/19.9.E17 Encodings: TeX, pMML, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

where

19.9.18		$\displaystyle L$	$\displaystyle=(1/\sigma)\operatorname{arctanh}\left(\sigma\sin\phi\right),$
	$\sigma=\sqrt{(1+k^{2})/2}$ ,
		$\displaystyle U$	$\displaystyle=\tfrac{1}{2}\operatorname{arctanh}\left(\sin\phi\right)+\tfrac{1% }{2}k^{-1}\operatorname{arctanh}\left(k\sin\phi\right).$
ⓘ Symbols: $\operatorname{arctanh}\NVar{z}$ : inverse hyperbolic tangent function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $L$ : lower limit and $U$ : upper limit Permalink: http://dlmf.nist.gov/19.9.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.9(ii), §19.9 and Ch.19

Other inequalities for $F\left(\phi,k\right)$ can be obtained from inequalities for $R_{F}\left(x,y,z\right)$ given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).

19.8 Quadratic Transformations 19.10 Relations to Other Functions

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§19.9 Inequalities

Contents

§19.9(i) Complete Integrals

§19.9(ii) Incomplete Integrals