tango.math.Elliptic

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real ellipticF(real phi, real m) ¶

Incomplete elliptic integral of the first kind

Approximates the integral F(phi | m) = ∫₀^phi dt/ (sqrt( 1- m sin² t))

of amplitude phi and modulus m, using the arithmetic - geometric mean algorithm.

real ellipticE(real phi, real m) ¶

Incomplete elliptic integral of the second kind

Approximates the integral

E(phi | m) = ∫₀^phi sqrt( 1- m sin² t) dt

of amplitude phi and modulus m, using the arithmetic - geometric mean algorithm.

real ellipticKComplete(real x) ¶

Complete elliptic integral of the first kind

Approximates the integral

K(m) = ∫₀^π/2 dt/ (sqrt( 1- m sin² t))

where m = 1 - x, using the approximation

P(x) - log x Q(x).

The argument x is used rather than m so that the logarithmic singularity at x = 1 will be shifted to the origin; this preserves maximum accuracy.

x must be in the range 0 <= x <= 1

This is equivalent to ellipticF(PI_2, 1-x).

K(0) = π/2.

real ellipticEComplete(real x) ¶

Complete elliptic integral of the second kind

Approximates the integral

E(m) = ∫₀^π/2 sqrt( 1- m sin² t) dt

where m = 1 - x, using the approximation

P(x) - x log x Q(x).

Though there are no singularities, the argument m1 is used rather than m for compatibility with ellipticKComplete().

E(1) = 1; E(0) = π/2. m must be in the range 0 <= m <= 1.

real ellipticPi(real phi, real m, real n) ¶

Incomplete elliptic integral of the third kind

Approximates the integral

PI(n; phi | m) = ∫_t=0^phi dt/((1 - n sin²t) * sqrt( 1- m sin² t))

of amplitude phi, modulus m, and characteristic n using Gauss-Legendre quadrature. Note that ellipticPi(PI_2, m, 1) is infinite for any m.

real ellipticPiComplete(real m, real n) ¶

Complete elliptic integral of the third kind