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Elliptic Integrals

Source: R/elliptic_integrals.R
elliptic_integrals.Rd

Functions to compute various elliptic integrals, including Carlson's elliptic integrals and incomplete elliptic integrals.

Elliptic Integrals - Carlson Form

  • ellint_rf(x, y, z): Carlson's Elliptic Integral \(R_F\)

$$R_F(x, y, z) = \frac{1}{2}\int_{0}^\infty[(t+x)(t+y)(t+z)]^{-\frac{1}{2}}dt$$

  • ellint_rd(x, y, z): Carlson's Elliptic Integral \(R_D\)

$$R_D(x, y, z) = \frac{3}{2}\int_{0}^\infty[(t+x)(t+y)]^{-\frac{1}{2}}(t+z)^{-\frac{3}{2}}dt$$

  • ellint_rj(x, y, z, p): Carlson's Elliptic Integral \(R_J\)

$$R_J(x, y, z) = \frac{3}{2}\int_{0}^\infty(t+p)^{-1}[(t+x)(t+y)(t+z)]^{-\frac{1}{2}}dt$$

  • ellint_rc(x, y): Carlson's Elliptic Integral \(R_C\)

$$R_C(x, y) = \frac{1}{2}\int_{0}^\infty(t+x)^{-\frac{1}{2}}(t+y)^{-1}dt$$

  • ellint_rg(x, y, z): Carlson's Elliptic Integral \(R_G\)

$$R_G(x, y, z) = \frac{1}{4\pi}\int_{0}^{2\pi}\int_{0}^{\pi}\sqrt{\left(x\sin^2\theta\cos^2\phi+y\sin^2\theta\sin^2\phi + z\cos^2\theta\right)}\sin\theta d\theta \phi$$

Elliptic Integrals of the First Kind - Legendre Form

  • ellint_1(k, phi): Incomplete elliptic integral of the first kind: \(F(\phi, k)\):

$$F(\phi, k) = \int_0^{\phi}\frac{d\theta}{\sqrt{1-k^2\sin^2\theta}}d\theta$$

Elliptic Integrals of the Second Kind - Legendre Form

  • ellint_2(k, phi): Incomplete elliptic integral of the second kind: \(E(\phi, k)\):

$$E(\phi, k) = \int_0^{\phi}\sqrt{1-k^2\sin^2\theta}d\theta$$

Elliptic Integrals of the Third Kind - Legendre Form

  • ellint_3(k, n, phi): Incomplete elliptic integral of the third kind: \(\Pi(n, \phi, k)\):

$$\Pi(n, \phi, k) = \int_0^{\phi}\frac{d\theta}{\left(1-n\sin^2\theta\right)\sqrt{1-k^2\sin^2\theta}}d\theta$$

Elliptic Integral D - Legendre Form

  • ellint_d(k, phi): Incomplete elliptic integral \(D(\phi, k)\):

$$D(\phi, k) = \frac{(F(\phi, k) - E(\phi, k))}{k^2}$$

Jacobi Zeta Function

  • jacobi_zeta(k, phi): Jacobi Zeta function \(Z(\phi, k)\):

$$Z(\phi, k) = E(\phi, k) - \frac{E(\frac{\pi}{2},k)F(\phi,k)}{F(\frac{\pi}{2}, k)}$$

Heuman Lambda Function

  • heuman_lambda(k, phi): Heuman Lambda function \(\Lambda_0(\phi, k)\):

$$\Lambda_0(\phi, k) = \frac{F(\phi,\sqrt{1-k^2})}{F(\frac{\pi}{2}, \sqrt{1-k^2})} + \frac{2}{\pi}F(\frac{\pi}{2},k)Z(\phi, \sqrt{1-k^2})$$

Usage

ellint_rf(x, y, z)

ellint_rd(x, y, z)

ellint_rj(x, y, z, p)

ellint_rc(x, y)

ellint_rg(x, y, z)

ellint_1(k, phi = NULL)

ellint_2(k, phi = NULL)

ellint_3(k, n, phi = NULL)

ellint_d(k, phi = NULL)

jacobi_zeta(k, phi)

heuman_lambda(k, phi)

Arguments

x

First parameter of Carlson's integral (must be non-negative)

y

Second parameter of Carlson's integral

z

Third parameter of Carlson's integral

p

Fourth parameter of the integral (for Rj, must be non-zero)

k

Elliptic modulus for Legendre-form integrals

phi

Amplitude (angle) for incomplete elliptic integrals

n

Characteristic for elliptic integrals of the third kind

Value

A single numeric value with the computed elliptic integral.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Carlson's elliptic integral Rf with parameters x = 1, y = 2, z = 3
ellint_rf(1, 2, 3)
#> [1] 0.7269459
# Carlson's elliptic integral Rd with parameters x = 1, y = 2, z = 3
ellint_rd(1, 2, 3)
#> [1] 0.2904603
# Carlson's elliptic integral Rj with parameters x = 1, y = 2, z = 3, p = 4
ellint_rj(1, 2, 3, 4)
#> [1] 0.2398481
# Carlson's elliptic integral Rc with parameters x = 1, y = 2
ellint_rc(1, 2)
#> [1] 0.7853982
# Carlson's elliptic integral Rg with parameters x = 1, y = 2, z = 3
ellint_rg(1, 2, 3)
#> [1] 1.401847
# Incomplete elliptic integral of the first kind with k = 0.5, phi = pi/4
ellint_1(0.5, pi / 4)
#> [1] 0.8043661
# Complete elliptic integral of the first kind
ellint_1(0.5)
#> [1] 1.68575
# Incomplete elliptic integral of the second kind with k = 0.5, phi = pi/4
ellint_2(0.5, pi / 4)
#> [1] 0.767196
# Complete elliptic integral of the second kind
ellint_2(0.5)
#> [1] 1.467462
# Incomplete elliptic integral of the third kind with k = 0.5, n = 0.5, phi = pi/4
ellint_3(0.5, 0.5, pi / 4)
#> [1] 0.8930657
# Complete elliptic integral of the third kind with k = 0.5, n = 0.5
ellint_3(0.5, 0.5)
#> [1] 2.413672
# Incomplete elliptic integral D with k = 0.5, phi = pi/4
ellint_d(0.5, pi / 4)
#> [1] 0.1486805
# Complete elliptic integral D
ellint_d(0.5)
#> [1] 0.8731526
# Jacobi zeta function with k = 0.5, phi = pi/4
jacobi_zeta(0.5, pi / 4)
#> [1] 0.06698741
# Heuman's lambda function with k = 0.5, phi = pi/4
heuman_lambda(0.5, pi / 4)
#> [1] 0.6632254