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19 Elliptic IntegralsLegendre’s Integrals

§19.6 Special Cases

  1. §19.6(i) Complete Elliptic Integrals
  2. §19.6(ii) F(ϕ,k)
  3. §19.6(iii) E(ϕ,k)
  4. §19.6(iv) Π(ϕ,α2,k)
  5. §19.6(v) RC(x,y)

§19.6(i) Complete Elliptic Integrals

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(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=1k2sin2ϕ. 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(c1,cα2),
Π(ϕ,α2,1) =11α2(RC(c,c1)α2RC(c,cα2)),
Π(ϕ,1,1) =12(RC(c,c1)+c(c1)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) =x1/2,
RC(λx,λy) =λ1/2RC(x,y),
RC(x,y) +,
y0+ or y0, x>0,
RC(0,y) =12πy1/2,
RC(0,y) =0,