Package xal.tools.math

Class EllipticIntegral

java.lang.Object

xal.tools.math.EllipticIntegral

public class EllipticIntegral
extends Object

Utility class for numerical computation of elliptic integrals.

Since:

Aug 17, 2006

Author:

Christopher K. Allen

Constructor Summary

Constructors

Constructor	Description
EllipticIntegral()

Method Summary

Modifier and Type	Method	Description
static double	formFactorD(double s)	Evaluates the value of the "form factor" which is an analytic means for approximating the Carlson elliptic integral RD(x,y,z).
static void	main(String[] args)	Testing Driver
static double	rd(double x, double y, double z)	Compute and return the Carlson Elliptic integral RD(x,y,z).
static double	symmetricRDr(double r, double z)	Axis-symmetric Carlson elliptic integral RD(z,r,r) (permuted arguments) Compute and return the Carlson elliptic integral for the "axis-symmetric" situation where x = y = r.
static double	symmetricRDz(double r, double z)	Axis-symmetric Carlson elliptic integral RD(r,r,z) Compute and return the Carlson elliptic integral for the "axis-symmetric" situation where x = y = r.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

EllipticIntegral

public EllipticIntegral()

Method Details

rd

public static double rd(double x,
 double y,
 double z)

Compute and return the Carlson Elliptic integral RD(x,y,z). This is the "degenerate form" of Carlson's Elliptic integral of the third kind, a special case of his elliptic integral RJ(x,y,z,p) where p == z. The definition of this function is RD(x,y,z) := (3/2)Integral_dt{ 1.0/( (t+x)^1/2*(t+y)^1/2*(t+z)^3/2 ) } where the integral is from zero to infinity.

Parameters:

x - real number > 0

y - real number > 0

z - real number > 0

Returns:

the value of RD(x,y,z)

symmetricRDz

public static double symmetricRDz(double r,
 double z)
                           throws IllegalArgumentException

Axis-symmetric Carlson elliptic integral RD(r,r,z) Compute and return the Carlson elliptic integral for the "axis-symmetric" situation where x = y = r. In this case there is an analytic formula for the integral involving inverse cosines and inverse hyperbolic cosines. The value of this function is actually computed by a call to the EllipticIntegral.formFactorD() function. The definition for symmetricRDz(r,z) is given by the following RDz(r,z) := (3/2)Integral_dt{ 1.0/( (t+r)*(t+z)^3/2 ) } = (3/ (r*z^1/2))*formFactorD( (z/r)^1/2 ) where the integral is from zero to infinity. NOTE: Depending upon the algorithm used to compute the Math.acos() and ElementaryFunction.acosh() functions it may actually be faster to use the function EllipticIntegral.RD(r,r,z).

Parameters:

r - real number > 0 (i.e., radius)

z - real number > 0 (i.e., length)

Returns:

value of RD(r,r,z)

Throws:

IllegalArgumentException - argument less than zero

See Also:

• rd(double, double, double)

symmetricRDr

public static double symmetricRDr(double r,
 double z)
                           throws IllegalArgumentException

Axis-symmetric Carlson elliptic integral RD(z,r,r) (permuted arguments) Compute and return the Carlson elliptic integral for the "axis-symmetric" situation where x = y = r. In this case there is an analytic formula for the integral involving inverse cosines and inverse hyperbolic cosines. The value of this function is actually computed by a call to the EllipticIntegral.formFactorD() function. The definition for symmetricRDr(z,r) is given by the following RDr(r,z) := (3/2)Integral_dt{ 1.0/( (t+z)^1/2*(t+r)^2 ) } = (1/2)*(3/(r*z^1/2) - *RD(r,r,z) where the integral is from zero to infinity. NOTE: Depending upon the algorithm used to compute the Math.acos() and ElementaryFunction.acosh() functions it may actually be faster to use the function EllipticIntegral.RD(r,r,z).

Parameters:

r - real number > 0 (i.e., radius)

z - real number > 0 (i.e., length)

Returns:

value of RD(z,r,r)

Throws:

IllegalArgumentException - argument less than zero

See Also:

• rd(double, double, double)
• symmetricRDz(double, double)

formFactorD

public static double formFactorD(double s)

Evaluates the value of the "form factor" which is an analytic means for approximating the Carlson elliptic integral RD(x,y,z). I believe this notion was introduced by Lapostalle in the early 1970s. I have the definition of the form factor f(s) as f(s) := (s/2)Integral_dt{1/( (t+1)*(t+s^2)^3/2 ) } where the integral is from zero to infinity.

Parameters:

s - a real number in the interval (0,inf)

Returns:

value of the form factor at s

main

public static void main(String[] args)

Testing Driver