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);
}

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 

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

[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

[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

[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

[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