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19 Elliptic IntegralsLegendre’s Integrals

§19.5 Maclaurin and Related Expansions

If |k|<1 and |α|<1, then

19.5.1 K(k)=π2m=0(12)m(12)mm!m!k2m=π2F12(12,121;k2),

where F12 is the Gauss hypergeometric function (§§15.1 and 15.2(i)).

19.5.2 E(k)=π2m=0(12)m(12)mm!m!k2m=π2F12(12,121;k2),
19.5.3 D(k)=π4m=0(32)m(12)m(m+1)!m!k2m=π4F12(32,122;k2),
19.5.4 Π(α2,k)=π2n=0(12)nn!m=0n(12)mm!k2mα2n2m=π2F1(12;12,1;1;k2,α2),
19.5.4_1 F(ϕ,k)=m=0(12)msin2m+1ϕ(2m+1)m!F12(m+12,12m+32;sin2ϕ)k2m=sinϕF1(12;12,12;32;sin2ϕ,k2sin2ϕ),
19.5.4_2 E(ϕ,k)=m=0(12)msin2m+1ϕ(2m+1)m!F12(m+12,12m+32;sin2ϕ)k2m=sinϕF1(12;12,12;32;sin2ϕ,k2sin2ϕ),
19.5.4_3 Π(ϕ,α2,k)=m=0(12)msin2m+1ϕ(2m+1)m!F1(m+12;12,1;m+32;sin2ϕ,α2sin2ϕ)k2m,

where F1(α;β,β;γ;x,y) is an Appell function (§16.13).

For Jacobi’s nome q:

19.5.5 q=exp(πK(k)/K(k))=r+8r2+84r3+992r4+,
r=116k2, 0k1.


19.5.6 q=λ+2λ5+15λ9+150λ13+1707λ17+,


19.5.7 λ=(1k)/(2(1+k)).

Coefficients of terms up to λ49 are given in Lee (1990), along with tables of fractional errors in K(k) and E(k), 0.1k20.9999, obtained by using 12 different truncations of (19.5.6) in (19.5.8) and (19.5.9).

19.5.8 K(k)=π2(1+2n=1qn2)2,
19.5.9 E(k)=K(k)+2π2K(k)n=1(1)nn2qn21+2n=1(1)nqn2,

An infinite series for lnK(k) is equivalent to the infinite product

19.5.10 K(k)=π2m=1(1+km),

where k0=k and

19.5.11 km+1=11km21+1km2,

Series expansions of F(ϕ,k) and E(ϕ,k) are surveyed and improved in Van de Vel (1969), and the case of F(ϕ,k) is summarized in Gautschi (1975, §1.3.2). For series expansions of Π(ϕ,α2,k) when |α2|<1 see Erdélyi et al. (1953b, §13.6(9)). See also Karp et al. (2007).