§19.9 Inequalities

Referenced by:: Ch.19
Permalink:: http://dlmf.nist.gov/19.9

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

§19.9(i) Complete Integrals

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).
Keywords:: Legendre’s elliptic integrals, inequalities
Referenced by:: §19.24(i), §19.30(i)
Permalink:: http://dlmf.nist.gov/19.9.i

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

19.9.1

$\mathop{\ln\/}\nolimits 4\leq\mathop{K\/}\nolimits\!\left(k\right)+\mathop{\ln\/}\nolimits k^{{\prime}}\leq\pi/2,$

$1\leq\mathop{E\/}\nolimits\!\left(k\right)\leq\pi/2.$

$1\leq(2/\pi)\sqrt{1-\alpha^{2}}\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)\leq 1/k^{{\prime}},$ $\alpha^{2}<1$ .

19.9.2 $1+\frac{{k^{{\prime}}}^{2}}{8}<\frac{\mathop{K\/}\nolimits\!\left(k\right)}{\mathop{\ln\/}\nolimits\!\left(4/k^{{\prime}}\right)}<1+\frac{{k^{{\prime}}}^{2}}{4},$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\ln\/}\nolimits z$ : 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
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19.9.3 $9+\frac{k^{2}{k^{{\prime}}}^{2}}{8}<\frac{(8+k^{2})\mathop{K\/}\nolimits\!\left(k\right)}{\mathop{\ln\/}\nolimits\!\left(4/k^{{\prime}}\right)}<9.096.$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\ln\/}\nolimits z$ : 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
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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 $\left(\frac{1+{k^{{\prime}}}^{{3/2}}}{2}\right)^{{2/3}}\leq\frac{2}{\pi}\mathop{E\/}\nolimits\!\left(k\right)\leq\left(\frac{1+{k^{{\prime}}}^{2}}{2}\right)^{{1/2}}$

Symbols:: $\mathop{E\/}\nolimits\!\left(k\right)$ : 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
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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 $\mathop{\ln\/}\nolimits\frac{(1+\sqrt{k^{{\prime}}})^{2}}{k}<\frac{\pi\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)}{2\!\mathop{K\/}\nolimits\!\left(k\right)}<\mathop{\ln\/}\nolimits\frac{2(1+k^{{\prime}})}{k}.$

Symbols:: $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\ln\/}\nolimits z$ : 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
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For a sharper, but more complicated, version of (19.9.5) see Anderson et al. (1990).

Other inequalities are:

19.9.6 $(1-\tfrac{3}{4}k^{2})^{{-1/2}}<\frac{4}{\pi k^{2}}(\mathop{K\/}\nolimits\!\left(k\right)-\mathop{E\/}\nolimits\!\left(k\right))<(k^{{\prime}})^{{-3/4}},$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : 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

19.9.7 $(1-\tfrac{1}{4}k^{2})^{{-1/2}}<\frac{4}{\pi k^{2}}(\mathop{E\/}\nolimits\!\left(k\right)-{k^{{\prime}}}^{2}\mathop{K\/}\nolimits\!\left(k\right))<\min((k^{{\prime}})^{{-1/4}},4/\pi),$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, $(a,b)$ : open interval, $k$ : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §19.9(i)
Permalink:: http://dlmf.nist.gov/19.9.E7
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19.9.8 $k^{{\prime}}<\frac{\mathop{E\/}\nolimits\!\left(k\right)}{\mathop{K\/}\nolimits\!\left(k\right)}<\left(\frac{1+k^{{\prime}}}{2}\right)^{2}.$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : 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

Further inequalities for $\mathop{K\/}\nolimits\!\left(k\right)$ and $\mathop{E\/}\nolimits\!\left(k\right)$ 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 $L(a,b)=4a\mathop{E\/}\nolimits\!\left(k\right),$ $k^{2}=1-(b^{2}/a^{2})$ , $a>b$ .

Symbols:: $\mathop{E\/}\nolimits\!\left(k\right)$ : 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
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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
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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

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)).
Keywords:: Legendre’s elliptic integrals, inequalities
Permalink:: http://dlmf.nist.gov/19.9.ii

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

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

19.9.11 $\phi\leq\mathop{F\/}\nolimits\!\left(\phi,k\right)\leq\min(\phi/\Delta,\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(\phi\right)),$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(x\right)$ : inverse Gudermannian function, $(a,b)$ : open interval, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$
Permalink:: http://dlmf.nist.gov/19.9.E11
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where $\mathop{{\mathrm{gd}^{{-1}}}\/}\nolimits\!\left(\phi\right)$ is given by (4.23.41) and (4.23.42). Also,

19.9.12 $\max(\mathop{\sin\/}\nolimits\phi,\phi\Delta)\leq\mathop{E\/}\nolimits\!\left(\phi,k\right)\leq\phi,$

Symbols:: $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $(a,b)$ : open interval, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $\Delta$
Permalink:: http://dlmf.nist.gov/19.9.E12
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19.9.13 $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},0\right)\leq\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)\leq\min(\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},0\right)/\Delta,\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},1\right)).$

Symbols:: $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $(a,b)$ : open interval, $\phi$ : 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

Sharper inequalities for $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ are:

19.9.14 $\frac{3}{1+\Delta+\mathop{\cos\/}\nolimits\phi}<\frac{\mathop{F\/}\nolimits\!\left(\phi,k\right)}{\mathop{\sin\/}\nolimits\phi}<\frac{1}{(\Delta\mathop{\cos\/}\nolimits\phi)^{{1/3}}},$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\sin\/}\nolimits z$ : 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
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19.9.15 $1<\mathop{F\/}\nolimits\!\left(\phi,k\right)\bigg/\left((\mathop{\sin\/}\nolimits\phi)\mathop{\ln\/}\nolimits\!\left(\frac{4}{\Delta+\mathop{\cos\/}\nolimits\phi}\right)\right)<\frac{4}{2+(1+k^{2}){\mathop{\sin\/}\nolimits^{{2}}}\phi}.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : 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
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19.9.16 $\mathop{F\/}\nolimits\!\left(\phi,k\right)=\frac{2}{\pi}\mathop{K\/}\nolimits\!\left(k^{{\prime}}\right)\mathop{\ln\/}\nolimits\!\left(\frac{4}{\Delta+\mathop{\cos\/}\nolimits\phi}\right)-\theta\Delta^{2},$ $(\mathop{\sin\/}\nolimits\phi)/8<\theta<(\mathop{\ln\/}\nolimits 2)/(k^{2}\mathop{\sin\/}\nolimits\phi)$ .

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : 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
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(19.9.15) is useful when $k^{2}$ and ${\mathop{\sin\/}\nolimits^{{2}}}\phi$ are both close to 1, since the bounds are then nearly equal; otherwise (19.9.14) is preferable.

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

19.9.17 $L\leq\mathop{F\/}\nolimits\!\left(\phi,k\right)\leq\sqrt{UL}\leq\tfrac{1}{2}(U+L)\leq U,$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : 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
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where

19.9.18

$L=(1/\sigma)\mathop{\mathrm{arctanh}\/}\nolimits\!\left(\sigma\mathop{\sin\/}\nolimits\phi\right),$ $\sigma=\sqrt{(1+k^{2})/2}$ ,

$U=\tfrac{1}{2}\mathop{\mathrm{arctanh}\/}\nolimits\!\left(\mathop{\sin\/}\nolimits\phi\right)+\tfrac{1}{2}k^{{-1}}\mathop{\mathrm{arctanh}\/}\nolimits\!\left(k\mathop{\sin\/}\nolimits\phi\right).$

Symbols:: $\mathop{\mathrm{arctanh}\/}\nolimits z$ : inverse hyperbolic tangent function, $\mathop{\sin\/}\nolimits z$ : 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

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

§19.9 Inequalities

Contents

§19.9(i) Complete Integrals

§19.9(ii) Incomplete Integrals