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19 Elliptic IntegralsLegendre’s Integrals

§19.6 Special Cases


§19.6(i) Complete Elliptic Integrals

19.6.3 Π(α2,0)=π/(21-α2),Π(0,k)=K(k),
19.6.4 Π(α2,k) +,
Π(α2,k) sign(1-α2),

Exact values of K(k) and E(k) for various special values of k are given in Byrd and Friedman (1971, 111.10 and 111.11) and Cooper et al. (2006).

§19.6(ii) F(ϕ,k)

19.6.7 F(0,k) =0,
F(ϕ,0) =ϕ,
F(12π,1) =,
F(12π,k) =K(k),
limϕ0F(ϕ,k)/ϕ =1.
19.6.8 F(ϕ,1)=(sinϕ)RC(1,cos2ϕ)=gd-1(ϕ).

For the inverse Gudermannian function gd-1(ϕ) see §4.23(viii). Compare also (19.10.2).

§19.6(iii) E(ϕ,k)

§19.6(iv) Π(ϕ,α2,k)

Circular and hyperbolic cases, including Cauchy principal values, are unified by using RC(x,y). Let c=csc2ϕα2 and Δ=1-k2sin2ϕ. Then

19.6.11 Π(0,α2,k) =0,
Π(ϕ,0,0) =ϕ,
Π(ϕ,1,0) =tanϕ.
19.6.12 Π(ϕ,α2,0) =RC(c-1,c-α2),
Π(ϕ,α2,1) =11-α2(RC(c,c-1)-α2RC(c,c-α2)),
Π(ϕ,1,1) =12(RC(c,c-1)+c(c-1)-1).
19.6.13 Π(ϕ,0,k) =F(ϕ,k),
Π(ϕ,k2,k) =1k2(E(ϕ,k)-k2Δsinϕcosϕ),
Π(ϕ,1,k) =F(ϕ,k)-1k2(E(ϕ,k)-Δtanϕ).
19.6.14 Π(12π,α2,k) =Π(α2,k),
limϕ0Π(ϕ,α2,k)/ϕ =1.

For the Cauchy principal value of Π(ϕ,α2,k) when α2>c, see §19.7(iii).

§19.6(v) RC(x,y)

19.6.15 RC(x,x) =x-1/2,
RC(λx,λy) =λ-1/2RC(x,y),
RC(x,y) +,
y0+ or y0-, x>0,
RC(0,y) =12πy-1/2,
RC(0,y) =0,