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11 Elliptic Integrals
• Carlson’s elliptic integral RC(x, y), Carlson’s elliptic integral of first
kind RF (x, y, z), Carlson’s symmetrized elliptic integral of second kind
RG(x, y, z), Carlson’s elliptic integral of second kind RD(x, y, z), and
Carlson’s elliptic integral of third kind RJ (x, y, z, p);
• Bulirsch’s elliptic integrals el1(x, kc), el2(x, kc, a, b), el3(x, kc, p), and
cel(kc, p, a, b); and
• Jacobi’s zeta function Z(β, k) and Heumann’s Λ function Λ(β, k).
All arguments could be complex.
ellipIntQuadtest The SPECFUNPHYS function res = ellipIntQuadTest
(whichEI, x,y,z, p) serves as test function for computing the elliptic
integrals F (φ, k), E(φ, k), Π(φ, n, k), RC(x, y), RF (x, y, z), RD(x, y, z), and
RJ (x, y, z, p) and is based on a direct quadrature.
11.2 Equations and Computation
The relevant equations useful for evaluating elliptic integrals, together with some
computational hints, will be listed in this section. The arguments k, φ, n, x, · · ·
could be complex numbers with the exception of real negative values. Thus, for
Legendre’s elliptic integrals (1 − sin
2 φ) ∈ C\] − ∞, 0] and (1 − k 2 sin
2 φ) ∈
C\] − ∞, 0] except that one could be zero, and (1 − n sin
2 φ) ∈ C \ {0}. The
computation of Legendre’s elliptic integrals and Bulirsch’s elliptic integrals will be
mainly based on Carlson’s elliptic integrals [1].
Legendre’s Elliptic Integral of First Kind K and F
The complete elliptic integral K(k) equals π/2 for k = 0 and has a pole for k = 1.
The computation is based on the product ansatz (10.12). In case of no convergency,
the evaluation will be based on the Carlson function RF
K(k) = RF (1 − k
2 , 1, 0).
(11.1a)
The incomplete elliptic integral F (φ, k) holds F (0, k) = 0, F (φ, 0) = φ, and
F (π/2, 1) = ∞; for all other cases, the computation is based on
F (φ, k) = sin φ · RF (cos
2 φ, 1 − k
2 sin
2 φ, 1).
(11.1b)
Legendre’s Elliptic Integral of Second Kind E
The incomplete elliptic integral of second kind is given by
E(φ, k) =
sin φ
0
(1 − k 2 t 2 )
(1 − t 2 )
dt
(11.2a)