Elliptic integral and function

Elliptic integral and function

[Liam Healy]

[-- Attachment #1: message body text --]
[-- Type: text/plain, Size: 524 bytes --]

My understanding is that the Jacobi elliptic function is the inverse
of the elliptic function.  That is, 
  sn(K(k),k) = 1
  cn(K(k),k) = 0
  dn(K(k),k) = sqrt(1-k^2)
see http://mathworld.wolfram.com/JacobiEllipticFunctions.html

but when I try this I get e.g.
gsltest 0.5
k=  0.50000000000000 K(k)=  1.68575035481260
sn=  0.99289175131682 cn=  0.11902088122262 dn=  0.71209759519570
all three of these seem wrong.  My driver program is attached.

What is wrong here?  Thank you for any light you can shed on this. 

Liam 


[-- Attachment #2: C source code to drive elliptic integrals and functions. --]
[-- Type: text/plain, Size: 962 bytes --]

/* ******************************************************** */
/*  file:        gsltest.c                                  */
/*  description: Test the GSL                               */
/*  date:        Mon Dec 17 2001 - 10:09                    */
/*  author:      Liam Healy <Liam.Healy@nrl.navy.mil>       */
/*  modified:    Mon Dec 17 2001 - 11:45 */
/* ******************************************************** */

/* Compile with:
   gcc gsltest.c -lgslcblas -lgsl -o gsltest 
   Run:
     source ~/bin/libpath /usr/local/lib/
     gsltest
*/


#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_sf_ellint.h>
#include <gsl/gsl_sf_elljac.h>

int main(int argc, char *argv[]) {
  double k, kk;
  double sn, cn, dn;
  int ret;

  k = atof(argv[1]);
  kk = gsl_sf_ellint_Kcomp (k, GSL_PREC_DOUBLE);
  ret = gsl_sf_elljac_e (kk, k, &sn, &cn, &dn); 

  printf("k=%18.14f K(k)=%18.14f\n",k,kk);
  printf("sn=%18.14f cn=%18.14f dn=%18.14f\n",sn,cn,dn);
}

Re: Elliptic integral and function

[Brian Gough]

Liam Healy writes:
 > My understanding is that the Jacobi elliptic function is the inverse
 > of the elliptic function.  That is, 
 >   sn(K(k),k) = 1
 >   cn(K(k),k) = 0
 >   dn(K(k),k) = sqrt(1-k^2)
 > see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
 > 

Hi,
Using the conventions in the GSL manual the relation is,

 sn(K(k),k^2) = 1
 cn(K(k),k^2) = 0
 dn(K(k),k^2) = sqrt(1-k^2)

which should work correctly. I think there is a note about the
different notations used by Carlson and Abramowitz&Stegun somewhere in
the chapter there.

regards

-- 
Brian Gough

----------------------------------------------------------------------
Network Theory Ltd            Phone: +44 (0)117 3179309
15 Royal Park                   WWW: http://www.network-theory.co.uk/
Clifton                       Email: bjg@network-theory.co.uk    
Bristol BS8 3AL
----------------------------------------------------------------------

Re: Elliptic integral and function

[Liam Healy]

>>>>> "Brian" == Brian Gough <bjg@network-theory.co.uk> writes:

    Brian> Liam Healy writes:
    >> My understanding is that the Jacobi elliptic function is the inverse
    >> of the elliptic function.  That is, 
    >> sn(K(k),k) = 1
    >> cn(K(k),k) = 0
    >> dn(K(k),k) = sqrt(1-k^2)
    >> see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
    >> 

    Brian> Hi,
    Brian> Using the conventions in the GSL manual the relation is,

    Brian>  sn(K(k),k^2) = 1
    Brian>  cn(K(k),k^2) = 0
    Brian>  dn(K(k),k^2) = sqrt(1-k^2)

    Brian> which should work correctly. I think there is a note about the
    Brian> different notations used by Carlson and Abramowitz&Stegun somewhere in
    Brian> the chapter there.

You're absolutely right, I had overlooked the m (where m=k^2).  And it
is documented, if a bit obscurely, "The Jacobian Elliptic functions are
defined in Abramowitz & Stegun, Chapter 16." so one has to hunt down
A&S for the definition and see how they've defined the arguments.

Thank you for solving this mystery.

Liam

Elliptic integral and function

[Liam Healy]

My understanding is that the Jacobi elliptic function is the inverse
of the elliptic function.  That is, 
  sn(K(k),k) = 1
  cn(K(k),k) = 0
  dn(K(k),k) = sqrt(1-k^2)
see http://mathworld.wolfram.com/JacobiEllipticFunctions.html

but when I try this I get e.g.
gsltest 0.5
k=  0.50000000000000 K(k)=  1.68575035481260
sn=  0.99289175131682 cn=  0.11902088122262 dn=  0.71209759519570
all three of these seem wrong.  My driver program is attached.

What is wrong here?  Thank you for any light you can shed on this. 

Liam 

Re: Elliptic integral and function

[Brian Gough]

Liam Healy writes:
 > My understanding is that the Jacobi elliptic function is the inverse
 > of the elliptic function.  That is, 
 >   sn(K(k),k) = 1
 >   cn(K(k),k) = 0
 >   dn(K(k),k) = sqrt(1-k^2)
 > see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
 > 

Hi,
Using the conventions in the GSL manual the relation is,

 sn(K(k),k^2) = 1
 cn(K(k),k^2) = 0
 dn(K(k),k^2) = sqrt(1-k^2)

which should work correctly. I think there is a note about the
different notations used by Carlson and Abramowitz&Stegun somewhere in
the chapter there.

regards

-- 
Brian Gough

----------------------------------------------------------------------
Network Theory Ltd            Phone: +44 (0)117 3179309
15 Royal Park                   WWW: http://www.network-theory.co.uk/
Clifton                       Email: bjg@network-theory.co.uk    
Bristol BS8 3AL
----------------------------------------------------------------------

Re: Elliptic integral and function

[Liam Healy]

>>>>> "Brian" == Brian Gough <bjg@network-theory.co.uk> writes:

    Brian> Liam Healy writes:
    >> My understanding is that the Jacobi elliptic function is the inverse
    >> of the elliptic function.  That is, 
    >> sn(K(k),k) = 1
    >> cn(K(k),k) = 0
    >> dn(K(k),k) = sqrt(1-k^2)
    >> see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
    >> 

    Brian> Hi,
    Brian> Using the conventions in the GSL manual the relation is,

    Brian>  sn(K(k),k^2) = 1
    Brian>  cn(K(k),k^2) = 0
    Brian>  dn(K(k),k^2) = sqrt(1-k^2)

    Brian> which should work correctly. I think there is a note about the
    Brian> different notations used by Carlson and Abramowitz&Stegun somewhere in
    Brian> the chapter there.

You're absolutely right, I had overlooked the m (where m=k^2).  And it
is documented, if a bit obscurely, "The Jacobian Elliptic functions are
defined in Abramowitz & Stegun, Chapter 16." so one has to hunt down
A&S for the definition and see how they've defined the arguments.

Thank you for solving this mystery.

Liam

Re: Elliptic integral and function

[Brian Gough]

Liam Healy writes:
 > My understanding is that the Jacobi elliptic function is the inverse
 > of the elliptic function.  That is, 
 >   sn(K(k),k) = 1
 >   cn(K(k),k) = 0
 >   dn(K(k),k) = sqrt(1-k^2)
 > see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
 > 

Hi,
Using the conventions in the GSL manual the relation is,

 sn(K(k),k^2) = 1
 cn(K(k),k^2) = 0
 dn(K(k),k^2) = sqrt(1-k^2)

which should work correctly. I think there is a note about the
different notations used by Carlson and Abramowitz&Stegun somewhere in
the chapter there.

regards

-- 
Brian Gough

----------------------------------------------------------------------
Network Theory Ltd            Phone: +44 (0)117 3179309
15 Royal Park                   WWW: http://www.network-theory.co.uk/
Clifton                       Email: bjg@network-theory.co.uk    
Bristol BS8 3AL
----------------------------------------------------------------------

Re: Elliptic integral and function

[Liam Healy]

>>>>> "Brian" == Brian Gough <bjg@network-theory.co.uk> writes:

    Brian> Liam Healy writes:
    >> My understanding is that the Jacobi elliptic function is the inverse
    >> of the elliptic function.  That is, 
    >> sn(K(k),k) = 1
    >> cn(K(k),k) = 0
    >> dn(K(k),k) = sqrt(1-k^2)
    >> see http://mathworld.wolfram.com/JacobiEllipticFunctions.html
    >> 

    Brian> Hi,
    Brian> Using the conventions in the GSL manual the relation is,

    Brian>  sn(K(k),k^2) = 1
    Brian>  cn(K(k),k^2) = 0
    Brian>  dn(K(k),k^2) = sqrt(1-k^2)

    Brian> which should work correctly. I think there is a note about the
    Brian> different notations used by Carlson and Abramowitz&Stegun somewhere in
    Brian> the chapter there.

You're absolutely right, I had overlooked the m (where m=k^2).  And it
is documented, if a bit obscurely, "The Jacobian Elliptic functions are
defined in Abramowitz & Stegun, Chapter 16." so one has to hunt down
A&S for the definition and see how they've defined the arguments.

Thank you for solving this mystery.

Liam

Elliptic integral and function

[Liam Healy]

[-- Attachment #1: message body text --]
[-- Type: text/plain, Size: 524 bytes --]

My understanding is that the Jacobi elliptic function is the inverse
of the elliptic function.  That is, 
  sn(K(k),k) = 1
  cn(K(k),k) = 0
  dn(K(k),k) = sqrt(1-k^2)
see http://mathworld.wolfram.com/JacobiEllipticFunctions.html

but when I try this I get e.g.
gsltest 0.5
k=  0.50000000000000 K(k)=  1.68575035481260
sn=  0.99289175131682 cn=  0.11902088122262 dn=  0.71209759519570
all three of these seem wrong.  My driver program is attached.

What is wrong here?  Thank you for any light you can shed on this. 

Liam 


[-- Attachment #2: C source code to drive elliptic integrals and functions. --]
[-- Type: text/plain, Size: 962 bytes --]

/* ******************************************************** */
/*  file:        gsltest.c                                  */
/*  description: Test the GSL                               */
/*  date:        Mon Dec 17 2001 - 10:09                    */
/*  author:      Liam Healy <Liam.Healy@nrl.navy.mil>       */
/*  modified:    Mon Dec 17 2001 - 11:45 */
/* ******************************************************** */

/* Compile with:
   gcc gsltest.c -lgslcblas -lgsl -o gsltest 
   Run:
     source ~/bin/libpath /usr/local/lib/
     gsltest
*/


#include <stdio.h>
#include <stdlib.h>
#include <gsl/gsl_sf_ellint.h>
#include <gsl/gsl_sf_elljac.h>

int main(int argc, char *argv[]) {
  double k, kk;
  double sn, cn, dn;
  int ret;

  k = atof(argv[1]);
  kk = gsl_sf_ellint_Kcomp (k, GSL_PREC_DOUBLE);
  ret = gsl_sf_elljac_e (kk, k, &sn, &cn, &dn); 

  printf("k=%18.14f K(k)=%18.14f\n",k,kk);
  printf("sn=%18.14f cn=%18.14f dn=%18.14f\n",sn,cn,dn);
}

Elliptic integral and function

[Liam Healy]

My understanding is that the Jacobi elliptic function is the inverse
of the elliptic function.  That is, 
  sn(K(k),k) = 1
  cn(K(k),k) = 0
  dn(K(k),k) = sqrt(1-k^2)
see http://mathworld.wolfram.com/JacobiEllipticFunctions.html

but when I try this I get e.g.
gsltest 0.5
k=  0.50000000000000 K(k)=  1.68575035481260
sn=  0.99289175131682 cn=  0.11902088122262 dn=  0.71209759519570
all three of these seem wrong.  My driver program is attached.

What is wrong here?  Thank you for any light you can shed on this. 

Liam