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19 Elliptic IntegralsLegendre’s Integrals

§19.12 Asymptotic Approximations

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

19.12.1 K(k)=m=0(12)m(12)mm!m!k2m(ln(1k)+d(m)),
0<|k|<1,
19.12.2 E(k)=1+12m=0(12)m(32)m(2)mm!k2m+2(ln(1k)+d(m)-1(2m+1)(2m+2)),
|k|<1,

where

19.12.3 d(m) =ψ(1+m)-ψ(12+m),
d(m+1) =d(m)-2(2m+1)(2m+2),
m=0,1,,

with d(0)=2ln2.

For the asymptotic behavior of F(ϕ,k) and E(ϕ,k) as ϕ12π- and k1- see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007).

Asymptotic approximations for Π(ϕ,α2,k), with different variables, are given in Karp et al. (2007). They are useful primarily when (1-k)/(1-sinϕ) is either small or large compared with 1.

If x0 and y>0, then

19.12.6 RC(x,y)=π2y-xy(1+O(xy)),
x/y0,
19.12.7 RC(x,y)=12x((1+y2x)ln(4xy)-y2x)(1+O(y2/x2)),
y/x0.