java.lang.Object
- gov.nih.mipav.model.algorithms.EllipticIntegral

```
public class EllipticIntegral
extends java.lang.Object
```
DOCUMENT ME!

Author:

ilb The elliptic integral of the first kind is defined by F(k, phi) = Integral from 0 to phi of d(theta)/sqrt(1 - k*k*sin(theta)*sin(theta)) where k is the modulus Letting t = sin(theta), the elliptic integral of the first kind can be written as F(k, x) = Integral from 0 to x of dt/sqrt((1-t*t)(1-k*k*t*t)) where x = sin(phi)
The elliptic integral of the second kind is defined by E(k, phi) = Integral from 0 to phi of sqrt(1 - k*k*sin(theta)*sin(theta))d(theta) where k is the modulus Letting t = sin(theta), the elliptic integral of the second kind can be written as E(k, x) = Integral from 0 to x of (sqrt(1 - k*k*t*t)/sqrt(1 - t*t))dt where x = sin(phi)

The elliptic integral of the third kind is defined by Pi(k, c, phi) = Integral from 0 to phi of d(theta)/((1 - c*sin(theta)*sin(theta))*sqrt(1 - k*k*sin(theta)*sin(theta))) Letting t = sin(theta), the elliptic integral of the third kind can be written as Pi(k, c, x) = Integral from 0 to x of dt/((1 - c*t*t)*sqrt((1 - t*t)*(1 - k*k*t*t))) where x = sin(phi)

When phi = PI/2 (x = 1), the elliptic integrals defined above are called complete elliptic integrals When phi is not equal to PI/2, the elliptic integrals are often referred to as incomplete elliptic integrals The complete elliptic integral of the first kind is: K(k) = Integral from 0 to PI/2 of d(theta)/sqrt(1 - k*k*sin(theta)*sin(theta)) where k is the modulus = Integral from 0 to 1 of dt/sqrt((1-t*t)(1-k*k*t*t)) The complete elliptic integral of the second kind is: E(k, phi) = Integral from 0 to PI/2 of sqrt(1 - k*k*sin(theta)*sin(theta))d(theta) where k is the modulus = Integral from 0 to 1 of (sqrt(1 - k*k*t*t)/sqrt(1 - t*t))dt The comlplete elliptic integral of the third kind is: Pi(k, c, phi) = Integral from 0 to PI/2 of d(theta)/((1 - c*sin(theta)*sin(theta))*sqrt(1 - k*k*sin(theta)*sin(theta))) = Integral from 0 to 1 of dt/((1 - c*t*t)*sqrt((1 - t*t)*(1 - k*k*t*t)))

The code for EllipticIntegralType = COMPLETE1, COMPLETE2, INCOMPLETE1, and INCOMPLETE2 is ported from a C++ program in Figures 1 and 2 of "Numerical Calculation of the Elliptic Integrals of the First and Second Kinds with Complex Modulus" by Tohru Morita, Interdisciplinary Information Sciences, Vol. 6, No. 1, 2000, pp. 67-74.

How to doublecheck? Use: 1.) Port of FORTRAN code for complete and incomplete elliptic integrals of the first kind with real arguments from Computation of Special Functions by Shanjie Zhang and Jianming Jin, John Wiley & Sons, Inc., 1996, Chapter 18, Elliptic Integrals and Jacobian Elliptic Functions, pp. 654-679.

Test results: For complete elliptic integrals of the first and second kind with real arguments, the Morita arithmetic-geometric mean, Morita Carlson's, ZHANG comelp, and ZHANG elit all give the same answer.

For incomplete elliptic integrals of the first and second kind with real arguemnts, both Morita Carlson's and ZHANG elit give the same answer.
On 2016/04/09 0:05, Gandler, William (NIH/CIT) [E] wrote: > Dear Interdisciplinary Information Sciences: > I am a computer programmer at the National Institutes of Health in Bethesda, Maryland, USA. I would like to incorporate elliptic integral calculation routines into the MIPAV software package. These routines are in Figures 1 and 2 of "Numerical Calculation of the Elliptic Integrals of the First and Second Kind with Complex Modulus" by Tohru Morita, Interdisciplinary Information Sciences, Vol. 6, No. 1, 2000, pp. 67-74. Who should I ask for permission? > > Sincerely, > > William Gandler Dear Dr William Gandler, Thank you very much for your inquiry. You may use the calculation routines published in our journal with clear statement of attribution. Sincerely, Nobuaki Obata Editor-in-Chief IIS

Field Summary

Fields
Modifier and Type	Field	Description
`private double[]`	`aki`	DOCUMENT ME!
`private double[]`	`akr`	DOCUMENT ME!
`private boolean`	`analyticContinuationUsed`	If false, function used If true, analytic continuation of function used.
`private double`	`BIGd`	DOCUMENT ME!
`private double`	`c`	DOCUMENT ME!
`private double`	`C1`	private double BIG = 3.0E137; // 3.0E37.
`private double`	`C1d`	DOCUMENT ME!
`private double`	`C2`	DOCUMENT ME!
`private double`	`C2d`	DOCUMENT ME!
`private double`	`C3`	DOCUMENT ME!
`private double`	`C3d`	DOCUMENT ME!
`private double`	`C4`	DOCUMENT ME!
`private double`	`C4d`	DOCUMENT ME!
`private double`	`C5d`	DOCUMENT ME!
`private double`	`C6d`	DOCUMENT ME!
`private double[]`	`cki`	DOCUMENT ME!
`private double[]`	`ckr`	DOCUMENT ME!
`private boolean`	`complementaryModulusUsed`	k is the modulus k', the complementary modulus = sqrt(1 - k*k).
`private boolean`	`complete`	DOCUMENT ME!
`private int[]`	`errorFlag`	DOCUMENT ME!
`private double`	`ERRTOL`	DOCUMENT ME!
`private double`	`ERRTOLb`	DOCUMENT ME!
`private double[]`	`first`	DOCUMENT ME!
`private double[]`	`firsti`	DOCUMENT ME!
`private double[]`	`firstr`	DOCUMENT ME!
`private double`	`mod`	modulus, 0
`private double`	`modi`	Imaginary part of modulus or complementary modulus.
`private double`	`modr`	Real part of modulus or complementary modulus.
`private static int`	`MORITA_SOURCE`	F(k, phi) INCOMPLETE1 E(k, phi) INCOMPLETE2 K(k) COMPLETE1 E(k) COMPLETE2 Complex arguments for first and second complete and incomplete elliptic integrals.
`private int`	`nrep`	DOCUMENT ME!
`private double`	`phi`	DOCUMENT ME!
`private double`	`phii`	DOCUMENT ME!
`private double`	`phir`	For complete integrals phir = PI/2, phii = 0.
`private double`	`pihf`	DOCUMENT ME!
`private double[]`	`rdi`	DOCUMENT ME!
`private double[]`	`rdr`	DOCUMENT ME!
`private double`	`RERR`	DOCUMENT ME!
`private double[]`	`rfi`	DOCUMENT ME!
`private double[]`	`rfr`	DOCUMENT ME!
`private boolean`	`runcel12`	DOCUMENT ME!
`private boolean`	`runcomelp`	DOCUMENT ME!
`private boolean`	`runelit`	DOCUMENT ME!
`private double[]`	`second`	DOCUMENT ME!
`private double[]`	`secondi`	DOCUMENT ME!
`private double[]`	`secondr`	DOCUMENT ME!
`private int`	`sgnck`	DOCUMENT ME!
`private int`	`source`	DOCUMENT ME!
`private static int`	`TEST_SOURCE`	Compare outputs of different modules for self-testing.
`private double[]`	`third`	DOCUMENT ME!
`private double`	`TINY`	DOCUMENT ME!
`private double`	`TINYd`	DOCUMENT ME!
`private ViewUserInterface`	`UI`	DOCUMENT ME!
`private boolean`	`useStandardMethod`	DOCUMENT ME!
`private static int`	`ZHANG_SOURCE`	Real arguments for first and second complete and incomplete elliptic integrals.

Constructor Summary

Constructors
Constructor	Description
`EllipticIntegral()`	~ Constructors ---------------------------------------------------------------------------------------------------
`EllipticIntegral(boolean complete, double modr, double modi, boolean complementaryModulusUsed, boolean analyticContinuationUsed, double phir, double phii, double[] firstr, double[] firsti, double[] secondr, double[] secondi, boolean useStandardMethod, int[] errorFlag)`	When calculating the complete integrals K(k) and E(K), we can make complete true or false, except when abs(k')
`EllipticIntegral(double mod, double[] first, double[] second)`	Used for complete elliptic integrals of first and second kind with real modulus 0
`EllipticIntegral(double modr, double modi, boolean complementaryModulusUsed, boolean analyticContinuationUsed, double[] firstr, double[] firsti, double[] secondr, double[] secondi, int[] errorFlag)`	Constructor only for complete integrals with phi = PI/2.
`EllipticIntegral(double mod, double phi, double[] first, double[] second)`	Used for complete and incomplete elliptic integrals of first and second kind with real modulus and argument 0
`EllipticIntegral(double mod, double phi, double c, double[] third)`	Used for complete and incomplete elliptic integrals of the third kind with real modulus, real argument phi, and real parameter c 0

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method	Description
`private void`	`cel12(double ckr, double cki)`	Routine cel12 only calculates complete integrals of the first and second kind with complex arguments Based on the arithmetic-geometric mean procedure Ported from Tohru Morita article.
`private void`	`comelp(double mod)`	Ported from Computation of Special Functions by Shanjie Zhang and Jianming Jin Computes complete first and second elliptic integrals with a real modulus 0
`private double`	`cosh(double x)`	DOCUMENT ME!
`private void`	`elit(double mod, double phi)`	Ported from Computation of Special Functions by Shanjie Zhang and Jianming Jin Computes complete and incomplete first and second elliptic integrals with a real modulus and real argument 0
`private void`	`elit3(double mod, double phi, double c)`	Ported from Computation of Special Functions by Shanjie Zhang and Jianming Jin Computes complete and incomplete elliptic integrals of the third kind with a real modulus real argument, and real parameter 0
`private void`	`ell12(double phir, double phii, double akr, double aki, double ckr, double cki, int sgnck)`	Calculates complete and incomplete elliptic integrals of the first and second kind with complex arguments Ported from the Tohru Morita article.
`private void`	`rfrd(double xrtr, double xrti, double yrtr, double yrti)`	Ported from Tohru Morita article.
`private void`	`rfrdb(double xrtr, double xrti, double yrtr, double yrti)`	Ported from Tohru Morita article.
`void`	`run()`	DOCUMENT ME!
`private void`	`runTest()`	DOCUMENT ME!
`private double`	`sinh(double x)`	DOCUMENT ME!
`private void`	`sqrtc(double zinr, double zini, double[] zsqr, double[] zsqi)`	Ported from Tohru Morita article Performs square root of complex argument.
`private double`	`zabs(double zr, double zi)`	zabs computes the absolute value or magnitude of a double precision complex variable zr + j*zi.
`private void`	`zcos(double inr, double ini, double[] outr, double[] outi)`	DOCUMENT ME!
`private void`	`zdiv(double ar, double ai, double br, double bi, double[] cr, double[] ci)`	complex divide c = a/b.
`private void`	`zmlt(double ar, double ai, double br, double bi, double[] cr, double[] ci)`	complex multiply c = a * b.
`private void`	`zsin(double inr, double ini, double[] outr, double[] outi)`	DOCUMENT ME!

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Field Detail
  - MORITA_SOURCE
```
private static final int MORITA_SOURCE
```
    F(k, phi) INCOMPLETE1 E(k, phi) INCOMPLETE2 K(k) COMPLETE1 E(k) COMPLETE2 Complex arguments for first and second complete and incomplete elliptic integrals.
    
    See Also:
    
    Constant Field Values
  - ZHANG_SOURCE
```
private static final int ZHANG_SOURCE
```
    Real arguments for first and second complete and incomplete elliptic integrals.
    
    See Also:
    
    Constant Field Values
  - TEST_SOURCE
```
private static final int TEST_SOURCE
```
    Compare outputs of different modules for self-testing.
    
    See Also:
    
    Constant Field Values
  - aki
```
private double[] aki
```
    DOCUMENT ME!
  - akr
```
private double[] akr
```
    DOCUMENT ME!
  - analyticContinuationUsed
```
private boolean analyticContinuationUsed
```
    If false, function used If true, analytic continuation of function used.
  - BIGd
```
private double BIGd
```
    DOCUMENT ME!
  - c
```
private double c
```
    DOCUMENT ME!
  - C1
```
private double C1
```
    private double BIG = 3.0E137; // 3.0E37.
  - C1d
```
private double C1d
```
    DOCUMENT ME!
  - C2
```
private double C2
```
    DOCUMENT ME!
  - C2d
```
private double C2d
```
    DOCUMENT ME!
  - C3
```
private double C3
```
    DOCUMENT ME!
  - C3d
```
private double C3d
```
    DOCUMENT ME!
  - C4
```
private double C4
```
    DOCUMENT ME!
  - C4d
```
private double C4d
```
    DOCUMENT ME!
  - C5d
```
private double C5d
```
    DOCUMENT ME!
  - C6d
```
private double C6d
```
    DOCUMENT ME!
  - cki
```
private double[] cki
```
    DOCUMENT ME!
  - ckr
```
private double[] ckr
```
    DOCUMENT ME!
  - complementaryModulusUsed
```
private boolean complementaryModulusUsed
```
    k is the modulus k', the complementary modulus = sqrt(1 - k*k).
  - complete
```
private boolean complete
```
    DOCUMENT ME!
  - errorFlag
```
private int[] errorFlag
```
    DOCUMENT ME!
  - ERRTOL
```
private double ERRTOL
```
    DOCUMENT ME!
  - ERRTOLb
```
private double ERRTOLb
```
    DOCUMENT ME!
  - first
```
private double[] first
```
    DOCUMENT ME!
  - firsti
```
private double[] firsti
```
    DOCUMENT ME!
  - firstr
```
private double[] firstr
```
    DOCUMENT ME!
  - mod
```
private double mod
```
    modulus, 0
    
    modi
    
    private double modi
    
    Imaginary part of modulus or complementary modulus.
    
    modr
    
    private double modr
    
    Real part of modulus or complementary modulus.
    
    nrep
    
    private int nrep
    
    DOCUMENT ME!
    
    phi
    
    private double phi
    
    DOCUMENT ME!
    
    phii
    
    private double phii
    
    DOCUMENT ME!
    
    phir
    
    private double phir
    
    For complete integrals phir = PI/2, phii = 0.
    
    pihf
    
    private double pihf
    
    DOCUMENT ME!
    
    rdi
    
    private double[] rdi
    
    DOCUMENT ME!
    
    rdr
    
    private double[] rdr
    
    DOCUMENT ME!
    
    RERR
    
    private double RERR
    
    DOCUMENT ME!
    
    rfi
    
    private double[] rfi
    
    DOCUMENT ME!
    
    rfr
    
    private double[] rfr
    
    DOCUMENT ME!
    
    runcel12
    
    private boolean runcel12
    
    DOCUMENT ME!
    
    runcomelp
    
    private boolean runcomelp
    
    DOCUMENT ME!
    
    runelit
    
    private boolean runelit
    
    DOCUMENT ME!
    
    second
    
    private double[] second
    
    DOCUMENT ME!
    
    secondi
    
    private double[] secondi
    
    DOCUMENT ME!
    
    secondr
    
    private double[] secondr
    
    DOCUMENT ME!
    
    sgnck
    
    private int sgnck
    
    DOCUMENT ME!
    
    source
    
    private int source
    
    DOCUMENT ME!
    
    third
    
    private double[] third
    
    DOCUMENT ME!
    
    TINY
    
    private double TINY
    
    DOCUMENT ME!
    
    TINYd
    
    private double TINYd
    
    DOCUMENT ME!
    
    UI
    
    private ViewUserInterface UI
    
    DOCUMENT ME!
    
    useStandardMethod
    
    private boolean useStandardMethod
    
    DOCUMENT ME!
    
    Constructor Detail
    
    EllipticIntegral
    
    public EllipticIntegral()
    
    ~ Constructors ---------------------------------------------------------------------------------------------------
    
    EllipticIntegral
    
    public EllipticIntegral(double mod, double[] first, double[] second)
    
    Used for complete elliptic integrals of first and second kind with real modulus 0
    
    Parameters:
    
    mod - DOCUMENT ME!
    
    first - DOCUMENT ME!
    
    second - DOCUMENT ME!
    
    EllipticIntegral
    
    public EllipticIntegral(double mod, double phi, double[] first, double[] second)
    
    Used for complete and incomplete elliptic integrals of first and second kind with real modulus and argument 0
    
    Parameters:
    
    mod - DOCUMENT ME!
    
    phi - DOCUMENT ME!
    
    first - DOCUMENT ME!
    
    second - DOCUMENT ME!
    
    EllipticIntegral
    
    public EllipticIntegral(double mod, double phi, double c, double[] third)
    
    Used for complete and incomplete elliptic integrals of the third kind with real modulus, real argument phi, and real parameter c 0
    
    Parameters:
    
    mod - DOCUMENT ME!
    
    phi - DOCUMENT ME!
    
    c - DOCUMENT ME!
    
    third - DOCUMENT ME!
    
    EllipticIntegral
    
    public EllipticIntegral(double modr, double modi, boolean complementaryModulusUsed, boolean analyticContinuationUsed, double[] firstr, double[] firsti, double[] secondr, double[] secondi, int[] errorFlag)
    
    Constructor only for complete integrals with phi = PI/2. The method called is based on the arithmetic-geometric mean procedure If only the complete integrals are calculated, the arithmetic-geometric mean procdure is superior to the 2 methods based on Carlson's algorithm
    
    Parameters:
    
    modr - DOCUMENT ME!
    
    modi - DOCUMENT ME!
    
    complementaryModulusUsed - DOCUMENT ME!
    
    analyticContinuationUsed - DOCUMENT ME!
    
    firstr - DOCUMENT ME!
    
    firsti - DOCUMENT ME!
    
    secondr - DOCUMENT ME!
    
    secondi - DOCUMENT ME!
    
    errorFlag - DOCUMENT ME!
    
    EllipticIntegral
    
    public EllipticIntegral(boolean complete, double modr, double modi, boolean complementaryModulusUsed, boolean analyticContinuationUsed, double phir, double phii, double[] firstr, double[] firsti, double[] secondr, double[] secondi, boolean useStandardMethod, int[] errorFlag)
    
    When calculating the complete integrals K(k) and E(K), we can make complete true or false, except when abs(k')
    
    Parameters:
    
    complete - DOCUMENT ME!
    
    modr - DOCUMENT ME!
    
    modi - DOCUMENT ME!
    
    complementaryModulusUsed - DOCUMENT ME!
    
    analyticContinuationUsed - DOCUMENT ME!
    
    phir - DOCUMENT ME!
    
    phii - DOCUMENT ME!
    
    firstr - DOCUMENT ME!
    
    firsti - DOCUMENT ME!
    
    secondr - DOCUMENT ME!
    
    secondi - DOCUMENT ME!
    
    useStandardMethod - DOCUMENT ME!
    
    errorFlag - DOCUMENT ME!
    
    Method Detail
    
    run
    
    public void run()
    
    DOCUMENT ME!
    
    cel12
    
    private void cel12(double ckr, double cki)
    
    Routine cel12 only calculates complete integrals of the first and second kind with complex arguments Based on the arithmetic-geometric mean procedure Ported from Tohru Morita article.
    
    Parameters:
    
    ckr - DOCUMENT ME!
    
    cki - DOCUMENT ME!
    
    comelp
    
    private void comelp(double mod)
    
    Ported from Computation of Special Functions by Shanjie Zhang and Jianming Jin Computes complete first and second elliptic integrals with a real modulus 0
    
    Parameters:
    
    mod - DOCUMENT ME!
    
    cosh
    
    private double cosh(double x)
    
    DOCUMENT ME!
    
    Parameters:
    
    x - DOCUMENT ME!
    
    Returns:
    
    DOCUMENT ME!
    
    elit
    
    private void elit(double mod, double phi)
    
    Ported from Computation of Special Functions by Shanjie Zhang and Jianming Jin Computes complete and incomplete first and second elliptic integrals with a real modulus and real argument 0
    
    Parameters:
    
    mod - DOCUMENT ME!
    
    phi - DOCUMENT ME!
    
    elit3
    
    private void elit3(double mod, double phi, double c)
    
    Ported from Computation of Special Functions by Shanjie Zhang and Jianming Jin Computes complete and incomplete elliptic integrals of the third kind with a real modulus real argument, and real parameter 0
    
    Parameters:
    
    mod - DOCUMENT ME!
    
    phi - DOCUMENT ME!
    
    c - DOCUMENT ME!
    
    ell12
    
    private void ell12(double phir, double phii, double akr, double aki, double ckr, double cki, int sgnck)
    
    Calculates complete and incomplete elliptic integrals of the first and second kind with complex arguments Ported from the Tohru Morita article.
    
    Parameters:
    
    phir - DOCUMENT ME!
    
    phii - DOCUMENT ME!
    
    akr - DOCUMENT ME!
    
    aki - DOCUMENT ME!
    
    ckr - DOCUMENT ME!
    
    cki - DOCUMENT ME!
    
    sgnck - DOCUMENT ME!
    
    rfrd
    
    private void rfrd(double xrtr, double xrti, double yrtr, double yrti)
    
    Ported from Tohru Morita article.
    
    Parameters:
    
    xrtr - DOCUMENT ME!
    
    xrti - DOCUMENT ME!
    
    yrtr - DOCUMENT ME!
    
    yrti - DOCUMENT ME!
    
    rfrdb
    
    private void rfrdb(double xrtr, double xrti, double yrtr, double yrti)
    
    Ported from Tohru Morita article.
    
    Parameters:
    
    xrtr - DOCUMENT ME!
    
    xrti - DOCUMENT ME!
    
    yrtr - DOCUMENT ME!
    
    yrti - DOCUMENT ME!
    
    runTest
    
    private void runTest()
    
    DOCUMENT ME!
    
    sinh
    
    private double sinh(double x)
    
    DOCUMENT ME!
    
    Parameters:
    
    x - DOCUMENT ME!
    
    Returns:
    
    DOCUMENT ME!
    
    sqrtc
    
    private void sqrtc(double zinr, double zini, double[] zsqr, double[] zsqi)
    
    Ported from Tohru Morita article Performs square root of complex argument.
    
    Parameters:
    
    zinr - DOCUMENT ME!
    
    zini - DOCUMENT ME!
    
    zsqr - DOCUMENT ME!
    
    zsqi - DOCUMENT ME!
    
    zabs
    
    private double zabs(double zr, double zi)
    
    zabs computes the absolute value or magnitude of a double precision complex variable zr + j*zi.
    
    Parameters:
    
    zr - double
    
    zi - double
    
    Returns:
    
    double
    
    zcos
    
    private void zcos(double inr, double ini, double[] outr, double[] outi)
    
    DOCUMENT ME!
    
    Parameters:
    
    inr - DOCUMENT ME!
    
    ini - DOCUMENT ME!
    
    outr - DOCUMENT ME!
    
    outi - DOCUMENT ME!
    
    zdiv
    
    private void zdiv(double ar, double ai, double br, double bi, double[] cr, double[] ci)
    
    complex divide c = a/b.
    
    Parameters:
    
    ar - double
    
    ai - double
    
    br - double
    
    bi - double
    
    cr - double[]
    
    ci - double[]
    
    zmlt
    
    private void zmlt(double ar, double ai, double br, double bi, double[] cr, double[] ci)
    
    complex multiply c = a * b.
    
    Parameters:
    
    ar - double
    
    ai - double
    
    br - double
    
    bi - double
    
    cr - double[]
    
    ci - double[]
    
    zsin
    
    private void zsin(double inr, double ini, double[] outr, double[] outi)
    
    DOCUMENT ME!
    
    Parameters:
    
    inr - DOCUMENT ME!
    
    ini - DOCUMENT ME!
    
    outr - DOCUMENT ME!
    
    outi - DOCUMENT ME!
    
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Class EllipticIntegral

Field Summary

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

MORITA_SOURCE

ZHANG_SOURCE

TEST_SOURCE

aki

akr

analyticContinuationUsed

BIGd

c

C1

C1d

C2

C2d

C3

C3d

C4

C4d

C5d

C6d

cki

ckr

complementaryModulusUsed

complete

errorFlag

ERRTOL

ERRTOLb

first

firsti

firstr

mod

modi

modr

nrep

phi

phii

phir

pihf

rdi

rdr

RERR

rfi

rfr

runcel12

runcomelp

runelit

second

secondi

secondr

sgnck

source

third

TINY

TINYd

UI

useStandardMethod

Constructor Detail

EllipticIntegral

EllipticIntegral

EllipticIntegral

EllipticIntegral

EllipticIntegral

EllipticIntegral

Method Detail

run

cel12

comelp

cosh

elit

elit3

ell12

rfrd

rfrdb

runTest

sinh

sqrtc