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19 Elliptic IntegralsLegendre’s Integrals

§19.9 Inequalities

  1. §19.9(i) Complete Integrals
  2. §19.9(ii) Incomplete Integrals

§19.9(i) Complete Integrals

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

19.9.1 ln4 K(k)+lnkπ/2,
1 E(k)π/2.
1 (2/π)1α2Π(α2,k)1/k,
19.9.2 1+k28<K(k)ln(4/k)<1+k24,
19.9.3 9+k2k28<(8+k2)K(k)ln(4/k)<9.096.

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<k20.922.

19.9.4 (1+k3/22)2/32πE(k)(1+k22)1/2

for 0k1. The lower bound in (19.9.4) is sharper than 2/π when 0k20.9960.

19.9.5 ln(1+k)2k<πK(k)2K(k)<ln2(1+k)k.

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

Other inequalities are:

19.9.6 (134k2)1/2<4πk2(K(k)E(k))<(k)3/4,
19.9.7 (114k2)1/2<4πk2(E(k)k2K(k))<min((k)1/4,4/π),
19.9.8 k<E(k)K(k)<(1+k2)2.

Further inequalities for K(k) and E(k) 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(k),
k2=1(b2/a2), a>b.

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)/(π(a+b)):

19.9.10 1+3λ210+43λ2+3λ10217,

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

Throughout this subsection we assume that 0<k<1, 0ϕπ/2, and Δ=1k2sin2ϕ>0.

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

19.9.11 ϕF(ϕ,k)min(ϕ/Δ,gd1(ϕ)),

where gd1(ϕ) is given by (4.23.41) and (4.23.42). Also,

19.9.12 max(sinϕ,ϕΔ)E(ϕ,k)ϕ,
19.9.13 Π(ϕ,α2,0)Π(ϕ,α2,k)min(Π(ϕ,α2,0)/Δ,Π(ϕ,α2,1)).

Sharper inequalities for F(ϕ,k) are:

19.9.14 31+Δ+cosϕ<F(ϕ,k)sinϕ<1(Δcosϕ)1/3,
19.9.15 1<F(ϕ,k)/((sinϕ)ln(4Δ+cosϕ))<42+(1+k2)sin2ϕ.
19.9.16 F(ϕ,k)=2πK(k)ln(4Δ+cosϕ)θΔ2,

(19.9.15) is useful when k2 and sin2ϕ are both close to 1, since the bounds are then nearly equal; otherwise (19.9.14) is preferable.

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

19.9.17 LF(ϕ,k)UL12(U+L)U,


19.9.18 L =(1/σ)arctanh(σsinϕ),
U =12arctanh(sinϕ)+12k1arctanh(ksinϕ).

Other inequalities for F(ϕ,k) can be obtained from inequalities for RF(x,y,z) given in Carlson (1966, (2.15)) and Carlson (1970) via (19.25.5).