Elliptic integrals and functions. More...

#include <EllipticFunction.hpp>

Public Member Functions
Constructor
	EllipticFunction (real k2=0, real alpha2=0)

	EllipticFunction (real k2, real alpha2, real kp2, real alphap2)

void	Reset (real k2=0, real alpha2=0)

void	Reset (real k2, real alpha2, real kp2, real alphap2)

Inspector functions.
Math::real	k2 () const

Math::real	kp2 () const

Math::real	alpha2 () const

Math::real	alphap2 () const

Complete elliptic integrals.
Math::real	K () const

Math::real	E () const

Math::real	D () const

Math::real	KE () const

Math::real	Pi () const

Math::real	G () const

Math::real	H () const

Incomplete elliptic integrals.
Math::real	F (real phi) const

Math::real	E (real phi) const

Math::real	Ed (real ang) const

Math::real	Einv (real x) const

Math::real	Pi (real phi) const

Math::real	D (real phi) const

Math::real	G (real phi) const

Math::real	H (real phi) const

Incomplete integrals in terms of Jacobi elliptic functions.
Math::real	F (real sn, real cn, real dn) const

Math::real	E (real sn, real cn, real dn) const

Math::real	Pi (real sn, real cn, real dn) const

Math::real	D (real sn, real cn, real dn) const

Math::real	G (real sn, real cn, real dn) const

Math::real	H (real sn, real cn, real dn) const

Periodic versions of incomplete elliptic integrals.
Math::real	deltaF (real sn, real cn, real dn) const

Math::real	deltaE (real sn, real cn, real dn) const

Math::real	deltaEinv (real stau, real ctau) const

Math::real	deltaPi (real sn, real cn, real dn) const

Math::real	deltaD (real sn, real cn, real dn) const

Math::real	deltaG (real sn, real cn, real dn) const

Math::real	deltaH (real sn, real cn, real dn) const

Elliptic functions.
void	sncndn (real x, real &sn, real &cn, real &dn) const

Math::real	Delta (real sn, real cn) const

Static Public Member Functions
Symmetric elliptic integrals.
static real	RF (real x, real y, real z)

static real	RF (real x, real y)

static real	RC (real x, real y)

static real	RG (real x, real y, real z)

static real	RG (real x, real y)

static real	RJ (real x, real y, real z, real p)

static real	RD (real x, real y, real z)

Private Types
enum	{ num_ = 13 }

typedef Math::real	real

Private Attributes
real	_alpha2

real	_alphap2

real	_Dc

real	_Ec

real	_eps

real	_Gc

real	_Hc

real	_k2

real	_Kc

real	_kp2

real	_Pic

Detailed Description

Elliptic integrals and functions.

This provides the elliptic functions and integrals needed for Ellipsoid, GeodesicExact, and TransverseMercatorExact. Two categories of function are provided:

static functions to compute symmetric elliptic integrals (http://dlmf.nist.gov/19.16.i)
member functions to compute Legrendre's elliptic integrals (http://dlmf.nist.gov/19.2.ii) and the Jacobi elliptic functions (http://dlmf.nist.gov/22.2).

In the latter case, an object is constructed giving the modulus k (and optionally the parameter α²). The modulus is always passed as its square k² which allows k to be pure imaginary (k² < 0). (Confusingly, Abramowitz and Stegun call m = k² the "parameter" and n = α² the "characteristic".)

In geodesic applications, it is convenient to separate the incomplete integrals into secular and periodic components, e.g.,

$E(\phi, k) = (2 E(k) / \pi) [ \phi + \delta E(\phi, k) ]$

where δE(φ, k) is an odd periodic function with period π.

The computation of the elliptic integrals uses the algorithms given in

B. C. Carlson, Computation of real or complex elliptic integrals, Numerical Algorithms 10, 13–26 (1995)

with the additional optimizations given in http://dlmf.nist.gov/19.36.i. The computation of the Jacobi elliptic functions uses the algorithm given in

R. Bulirsch, Numerical Calculation of Elliptic Integrals and Elliptic Functions, Numericshe Mathematik 7, 78–90 (1965).

The notation follows http://dlmf.nist.gov/19 and http://dlmf.nist.gov/22

Example of use:

// Example of using the GeographicLib::EllipticFunction class
#include <iostream>
#include <iomanip>
#include <exception>
#include <cmath>
#include <GeographicLib/Math.hpp>
#include <GeographicLib/EllipticFunction.hpp>
using namespace std;
using namespace GeographicLib;
int main() {
  try {
    EllipticFunction ell(0.1);  // parameter m = 0.1
    // See Abramowitz and Stegun, table 17.1
    cout << ell.K() << " " << ell.E() << "\n";
    double phi = 20, sn, cn;
    Math::sincosd(phi, sn ,cn);
    // See Abramowitz and Stegun, table 17.6 with
    // alpha = asin(sqrt(m)) = 18.43 deg and phi = 20 deg
    cout << ell.E(phi * Math::degree()) << " "
         << ell.E(sn, cn, ell.Delta(sn, cn))
         << "\n";
    // See Carlson 1995, Sec 3.
    cout << fixed << setprecision(16)
         << "RF(1,2,0)      = " << EllipticFunction::RF(1,2)      << "\n"
         << "RF(2,3,4)      = " << EllipticFunction::RF(2,3,4)    << "\n"
         << "RC(0,1/4)      = " << EllipticFunction::RC(0,0.25)   << "\n"
         << "RC(9/4,2)      = " << EllipticFunction::RC(2.25,2)   << "\n"
         << "RC(1/4,-2)     = " << EllipticFunction::RC(0.25,-2)  << "\n"
         << "RJ(0,1,2,3)    = " << EllipticFunction::RJ(0,1,2,3)  << "\n"
         << "RJ(2,3,4,5)    = " << EllipticFunction::RJ(2,3,4,5)  << "\n"
         << "RD(0,2,1)      = " << EllipticFunction::RD(0,2,1)    << "\n"
         << "RD(2,3,4)      = " << EllipticFunction::RD(2,3,4)    << "\n"
         << "RG(0,16,16)    = " << EllipticFunction::RG(16,16)    << "\n"
         << "RG(2,3,4)      = " << EllipticFunction::RG(2,3,4)    << "\n"
         << "RG(0,0.0796,4) = " << EllipticFunction::RG(0.0796,4) << "\n";
  }
  catch (const exception& e) {
    cout << "Caught exception: " << e.what() << "\n";
  }
}

Definition at line 62 of file EllipticFunction.hpp.

Member Typedef Documentation

typedef Math::real GeographicLib::EllipticFunction::real

private

Definition at line 64 of file EllipticFunction.hpp.

Member Enumeration Documentation

anonymous enum

private

Enumerator
num_

Definition at line 66 of file EllipticFunction.hpp.

Constructor & Destructor Documentation

GeographicLib::EllipticFunction::EllipticFunction	(	real	k2 = `0`,
		real	alpha2 = `0`
	)

inline

Constructor specifying the modulus and parameter.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (−∞, 1].
[in]	alpha2	the parameter α². α² must lie in (−∞, 1].

Exceptions

GeographicErr if k2 or alpha2 is out of its legal range.

If only elliptic integrals of the first and second kinds are needed, then set α² = 0 (the default value); in this case, we have Π(φ, 0, k) = F(φ, k), G(φ, 0, k) = E(φ, k), and H(φ, 0, k) = F(φ, k) - D(φ, k).

Definition at line 89 of file EllipticFunction.hpp.

GeographicLib::EllipticFunction::EllipticFunction	(	real	k2,
		real	alpha2,
		real	kp2,
		real	alphap2
	)

inline

Constructor specifying the modulus and parameter and their complements.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (−∞, 1].
[in]	alpha2	the parameter α². α² must lie in (−∞, 1].
[in]	kp2	the complementary modulus squared k'² = 1 − k². This must lie in [0, ∞).
[in]	alphap2	the complementary parameter α'² = 1 − α². This must lie in [0, ∞).

Exceptions

GeographicErr if k2, alpha2, kp2, or alphap2 is out of its legal range.

The arguments must satisfy k2 + kp2 = 1 and alpha2 + alphap2 = 1. (No checking is done that these conditions are met.) This constructor is provided to enable accuracy to be maintained, e.g., when k is very close to unity.

Definition at line 111 of file EllipticFunction.hpp.

Member Function Documentation

Math::real GeographicLib::EllipticFunction::alpha2 ( ) const

inline

Returns: the parameter α².

Definition at line 168 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::alphap2 ( ) const

inline

Returns: the complementary parameter α'² = 1 − α².

Definition at line 174 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::D ( ) const

inline

Jahnke's complete integral.

Returns: D(k).

D(k) is defined in http://dlmf.nist.gov/19.2.E6

$D(k) = \int_0^{\pi/2} \frac{\sin^2\phi}{\sqrt{1-k^2\sin^2\phi}}\,d\phi.$

Definition at line 215 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::D ( real phi ) const

Jahnke's incomplete elliptic integral.

Parameters

[in] phi

Returns: D(φ, k).

D(φ, k) is defined in http://dlmf.nist.gov/19.2.E4

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

Definition at line 520 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::D	(	real	sn,
		real	cn,
		real	dn
	)		const

Jahnke's incomplete elliptic integral in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: D(φ, k) as though φ ∈ (−π, π].

Definition at line 404 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::Delta	(	real	sn,
		real	cn
	)		const

inline

The Δ amplitude function.

Parameters

[in]	sn	sinφ.
[in]	cn	cosφ.

Returns: Δ = sqrt(1 − k² sin²φ).

Definition at line 584 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::deltaD	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic Jahnke's incomplete elliptic integral.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π D(φ, k) / (2 D(k)) - φ.

Definition at line 476 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::deltaE	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic incomplete integral of the second kind.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π E(φ, k) / (2 E(k)) - φ.

Definition at line 464 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::deltaEinv	(	real	stau,
		real	ctau
	)		const

The periodic inverse of the incomplete integral of the second kind.

Parameters

[in]	stau	= sinτ.
[in]	ctau	= sinτ.

Returns: the periodic function E⁻¹(τ (2 E(k)/π), k) - τ.

Definition at line 563 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::deltaF	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic incomplete integral of the first kind.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π F(φ, k) / (2 K(k)) - φ.

Definition at line 458 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::deltaG	(	real	sn,
		real	cn,
		real	dn
	)		const

Legendre's periodic geodesic longitude integral.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π G(φ, k) / (2 G(k)) - φ.

Definition at line 482 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::deltaH	(	real	sn,
		real	cn,
		real	dn
	)		const

Cayley's periodic geodesic longitude difference integral.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π H(φ, k) / (2 H(k)) - φ.

Definition at line 488 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::deltaPi	(	real	sn,
		real	cn,
		real	dn
	)		const

The periodic incomplete integral of the third kind.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: the periodic function π Π(φ, α², k) / (2 Π(α², k)) - φ.

Definition at line 470 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::E ( ) const

inline

The complete integral of the second kind.

Returns: E(k).

E(k) is defined in http://dlmf.nist.gov/19.2.E5

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

Definition at line 202 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::E ( real phi ) const

The incomplete integral of the second kind.

Parameters

[in] phi

Returns: E(φ, k).

E(φ, k) is defined in http://dlmf.nist.gov/19.2.E5

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

Definition at line 500 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::E	(	real	sn,
		real	cn,
		real	dn
	)		const

The incomplete integral of the second kind in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: E(φ, k) as though φ ∈ (−π, π].

Definition at line 381 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::Ed ( real ang ) const

The incomplete integral of the second kind with the argument given in degrees.

Parameters

[in] ang in degrees.

Returns: E(π ang/180, k).

Definition at line 506 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::Einv ( real x ) const

The inverse of the incomplete integral of the second kind.

Parameters

[in] x

Returns: φ = E⁻¹(x, k); i.e., the solution of such that E(φ, k) = x.

Definition at line 538 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::F ( real phi ) const

The incomplete integral of the first kind.

Parameters

[in] phi

Returns: F(φ, k).

F(φ, k) is defined in http://dlmf.nist.gov/19.2.E4

$F(\phi, k) = \int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}}\,d\theta.$

Definition at line 494 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::F	(	real	sn,
		real	cn,
		real	dn
	)		const

The incomplete integral of the first kind in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: F(φ, k) as though φ ∈ (−π, π].

Definition at line 370 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::G ( ) const

inline

Legendre's complete geodesic longitude integral.

Returns: G(α², k).

G(α², k) is given by

$G(\alpha^2, k) = \int_0^{\pi/2} \frac{\sqrt{1-k^2\sin^2\phi}}{1 - \alpha^2\sin^2\phi}\,d\phi.$

Definition at line 250 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::G ( real phi ) const

Legendre's geodesic longitude integral.

Parameters

[in] phi

Returns: G(φ, α², k).

G(φ, α², k) is defined by

$\begin{align} G(\phi, \alpha^2, k) &= \frac{k^2}{\alpha^2} F(\phi, k) + \biggl(1 - \frac{k^2}{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\ &= \int_0^\phi \frac{\sqrt{1-k^2\sin^2\theta}}{1 - \alpha^2\sin^2\theta}\,d\theta. \end{align}$

Legendre expresses the longitude of a point on the geodesic in terms of this combination of elliptic integrals in Exercices de Calcul Intégral, Vol. 1 (1811), p. 181, https://books.google.com/books?id=riIOAAAAQAAJ&pg=PA181.

See geodellip for the expression for the longitude in terms of this function.

Definition at line 526 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::G	(	real	sn,
		real	cn,
		real	dn
	)		const

Legendre's geodesic longitude integral in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: G(φ, α², k) as though φ ∈ (−π, π].

Definition at line 431 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::H ( ) const

inline

Cayley's complete geodesic longitude difference integral.

Returns: H(α², k).

H(α², k) is given by

$H(\alpha^2, k) = \int_0^{\pi/2} \frac{\cos^2\phi}{(1-\alpha^2\sin^2\phi)\sqrt{1-k^2\sin^2\phi}} \,d\phi.$

Definition at line 264 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::H ( real phi ) const

Cayley's geodesic longitude difference integral.

Parameters

[in] phi

Returns: H(φ, α², k).

H(φ, α², k) is defined by

$\begin{align} H(\phi, \alpha^2, k) &= \frac1{\alpha^2} F(\phi, k) + \biggl(1 - \frac1{\alpha^2}\biggr) \Pi(\phi, \alpha^2, k) \\ &= \int_0^\phi \frac{\cos^2\theta} {(1-\alpha^2\sin^2\theta)\sqrt{1-k^2\sin^2\theta}} \,d\theta. \end{align}$

Cayley expresses the longitude difference of a point on the geodesic in terms of this combination of elliptic integrals in Phil. Mag. 40 (1870), p. 333, https://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA333.

See geodellip for the expression for the longitude in terms of this function.

Definition at line 532 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::H	(	real	sn,
		real	cn,
		real	dn
	)		const

Cayley's geodesic longitude difference integral in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: H(φ, α², k) as though φ ∈ (−π, π].

Definition at line 444 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::K ( ) const

inline

The complete integral of the first kind.

Returns: K(k).

K(k) is defined in http://dlmf.nist.gov/19.2.E4

$K(k) = \int_0^{\pi/2} \frac1{\sqrt{1-k^2\sin^2\phi}}\,d\phi.$

Definition at line 190 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::k2 ( ) const

inline

Returns: the square of the modulus k².

Definition at line 157 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::KE ( ) const

inline

The difference between the complete integrals of the first and second kinds.

Returns: K(k) − E(k).

Definition at line 223 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::kp2 ( ) const

inline

Returns: the square of the complementary modulus k'² = 1 − k².

Definition at line 163 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::Pi ( ) const

inline

The complete integral of the third kind.

Returns: Π(α², k).

Π(α², k) is defined in http://dlmf.nist.gov/19.2.E7

$\Pi(\alpha^2, k) = \int_0^{\pi/2} \frac1{\sqrt{1-k^2\sin^2\phi}(1 - \alpha^2\sin^2\phi)}\,d\phi.$

Definition at line 237 of file EllipticFunction.hpp.

Math::real GeographicLib::EllipticFunction::Pi ( real phi ) const

The incomplete integral of the third kind.

Parameters

[in] phi

Returns: Π(φ, α², k).

Π(φ, α², k) is defined in http://dlmf.nist.gov/19.2.E7

$\Pi(\phi, \alpha^2, k) = \int_0^\phi \frac1{\sqrt{1-k^2\sin^2\theta}(1 - \alpha^2\sin^2\theta)}\,d\theta.$

Definition at line 514 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::Pi	(	real	sn,
		real	cn,
		real	dn
	)		const

The incomplete integral of the third kind in terms of Jacobi elliptic functions.

Parameters

[in]	sn	= sinφ.
[in]	cn	= cosφ.
[in]	dn	= sqrt(1 − k² sin²φ).

Returns: Π(φ, α², k) as though φ ∈ (−π, π].

Definition at line 416 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RC	(	real	x,
		real	y
	)

static

Degenerate symmetric integral of the first kind R_C.

Parameters

[in]	x
[in]	y

Returns: R_C(x, y) = R_F(x, y, y).

R_C is defined in http://dlmf.nist.gov/19.2.E17

$R_C(x, y) = \frac12 \int_0^\infty\frac1{\sqrt{t + x}(t + y)}\,dt$

Definition at line 81 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RD	(	real	x,
		real	y,
		real	z
	)

static

Degenerate symmetric integral of the third kind R_D.

Parameters

[in]	x
[in]	y
[in]	z

Returns: R_D(x, y, z) = R_J(x, y, z, z).

R_D is defined in http://dlmf.nist.gov/19.16.E5

$R_D(x, y, z) = \frac32 \int_0^\infty[(t + x) (t + y)]^{-1/2} (t + z)^{-3/2}\, dt$

Definition at line 180 of file src/EllipticFunction.cpp.

void GeographicLib::EllipticFunction::Reset	(	real	k2 = `0`,
		real	alpha2 = `0`
	)

inline

Reset the modulus and parameter.

Parameters

[in]	k2	the new value of square of the modulus k² which must lie in (−∞, ]. done.)
[in]	alpha2	the new value of parameter α². α² must lie in (−∞, 1].

Exceptions

GeographicErr if k2 or alpha2 is out of its legal range.

Definition at line 125 of file EllipticFunction.hpp.

void GeographicLib::EllipticFunction::Reset	(	real	k2,
		real	alpha2,
		real	kp2,
		real	alphap2
	)

Reset the modulus and parameter supplying also their complements.

Parameters

[in]	k2	the square of the modulus k². k² must lie in (−∞, 1].
[in]	alpha2	the parameter α². α² must lie in (−∞, 1].
[in]	kp2	the complementary modulus squared k'² = 1 − k². This must lie in [0, ∞).
[in]	alphap2	the complementary parameter α'² = 1 − α². This must lie in [0, ∞).

Exceptions

GeographicErr if k2, alpha2, kp2, or alphap2 is out of its legal range.

The arguments must satisfy k2 + kp2 = 1 and alpha2 + alphap2 = 1. (No checking is done that these conditions are met.) This constructor is provided to enable accuracy to be maintained, e.g., when is very small.

Definition at line 224 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RF	(	real	x,
		real	y,
		real	z
	)

static

Symmetric integral of the first kind R_F.

Parameters

[in]	x
[in]	y
[in]	z

Returns: R_F(x, y, z).

R_F is defined in http://dlmf.nist.gov/19.16.E1

$R_F(x, y, z) = \frac12 \int_0^\infty\frac1{\sqrt{(t + x) (t + y) (t + z)}}\, dt$

If one of the arguments is zero, it is more efficient to call the two-argument version of this function with the non-zero arguments.

Definition at line 29 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RF	(	real	x,
		real	y
	)

static

Complete symmetric integral of the first kind, R_F with one argument zero.

Parameters

[in]	x
[in]	y

Returns: R_F(x, y, 0).

Definition at line 66 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RG	(	real	x,
		real	y,
		real	z
	)

static

Symmetric integral of the second kind R_G.

Parameters

[in]	x
[in]	y
[in]	z

Returns: R_G(x, y, z).

R_G is defined in Carlson, eq 1.5

$R_G(x, y, z) = \frac14 \int_0^\infty[(t + x) (t + y) (t + z)]^{-1/2} \biggl( \frac x{t + x} + \frac y{t + y} + \frac z{t + z} \biggr)t\,dt$

See also http://dlmf.nist.gov/19.16.E3. If one of the arguments is zero, it is more efficient to call the two-argument version of this function with the non-zero arguments.

Definition at line 96 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RG	(	real	x,
		real	y
	)

static

Complete symmetric integral of the second kind, R_G with one argument zero.

Parameters

[in]	x
[in]	y

Returns: R_G(x, y, 0).

Definition at line 104 of file src/EllipticFunction.cpp.

Math::real GeographicLib::EllipticFunction::RJ	(	real	x,
		real	y,
		real	z,
		real	p
	)

static

Symmetric integral of the third kind R_J.

Parameters

[in]	x
[in]	y
[in]	z
[in]	p

Returns: R_J(x, y, z, p).

R_J is defined in http://dlmf.nist.gov/19.16.E2

$R_J(x, y, z, p) = \frac32 \int_0^\infty [(t + x) (t + y) (t + z)]^{-1/2} (t + p)^{-1}\, dt$

Definition at line 127 of file src/EllipticFunction.cpp.

void GeographicLib::EllipticFunction::sncndn	(	real	x,
		real &	sn,
		real &	cn,
		real &	dn
	)		const

The Jacobi elliptic functions.

Parameters

[in]	x	the argument.
[out]	sn	sn(x, k).
[out]	cn	cn(x, k).
[out]	dn	dn(x, k).

Definition at line 315 of file src/EllipticFunction.cpp.

Member Data Documentation

real GeographicLib::EllipticFunction::_alpha2

private

Definition at line 67 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_alphap2

private

Definition at line 67 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_Dc

private

Definition at line 68 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_Ec

private

Definition at line 68 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_eps

private

Definition at line 67 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_Gc

private

Definition at line 68 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_Hc

private

Definition at line 68 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_k2

private

Definition at line 67 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_Kc

private

Definition at line 68 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_kp2

private

Definition at line 67 of file EllipticFunction.hpp.

real GeographicLib::EllipticFunction::_Pic

private

Definition at line 68 of file EllipticFunction.hpp.

The documentation for this class was generated from the following files:

Public Member Functions

Static Public Member Functions

Private Types

Private Attributes

Detailed Description

Member Typedef Documentation

Member Enumeration Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation