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  • Committer: Bazaar Package Importer
  • Author(s): Camm Maguire
  • Date: 2004-11-13 18:39:14 UTC
  • mto: (2.1.2 hoary) (3.2.1 sid) (1.1.5 upstream)
  • mto: This revision was merged to the branch mainline in revision 3.
  • Revision ID: james.westby@ubuntu.com-20041113183914-ttig0evwuatnqosl
Tags: upstream-5.9.1
Import upstream version 5.9.1

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doc/info/Elliptic.texi
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Some examples of elliptic functions:
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@example
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(C1) jacobi_sn(u,m);
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(D1)                            JACOBI_SN(u, m)
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(C2) jacobi_sn(u,1);
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(D2)                                TANH(u)
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(C3) jacobi_sn(u,0);
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(D3)                                SIN(u)
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(C4) diff(jacobi_sn(u,m),u);
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(D4)                    JACOBI_CN(u, m) JACOBI_DN(u, m)
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(C5) diff(jacobi_sn(u,m),m);
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(D5) JACOBI_CN(u, m) JACOBI_DN(u, m)
 
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(%i1) jacobi_sn(u,m);
 
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(%o1)                            JACOBI_SN(u, m)
 
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(%i2) jacobi_sn(u,1);
 
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(%o2)                                TANH(u)
 
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(%i3) jacobi_sn(u,0);
 
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(%o3)                                SIN(u)
 
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(%i4) diff(jacobi_sn(u,m),u);
 
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(%o4)                    JACOBI_CN(u, m) JACOBI_DN(u, m)
 
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(%i5) diff(jacobi_sn(u,m),m);
 
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(%o5) JACOBI_CN(u, m) JACOBI_DN(u, m)
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      ELLIPTIC_E(ASIN(JACOBI_SN(u, m)), m)
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 (u - ------------------------------------)/(2 m)
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Some examples of elliptic integrals:
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@example
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(C1) elliptic_f(phi,m);
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(D1)                          ELLIPTIC_F(PHI, m)
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(C2) elliptic_f(phi,0);
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(D2)                                  PHI
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(C3) elliptic_f(phi,1);
 
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(%i1) elliptic_f(phi,m);
 
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(%o1)                          ELLIPTIC_F(PHI, m)
 
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(%i2) elliptic_f(phi,0);
 
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(%o2)                                  PHI
 
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(%i3) elliptic_f(phi,1);
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                                      PHI   %PI
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(D3)                          LOG(TAN(--- + ---))
 
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(%o3)                          LOG(TAN(--- + ---))
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                                       2     4
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(C4) elliptic_e(phi,1);
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(D4)                               SIN(PHI)
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(C5) elliptic_e(phi,0);
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(D5)                                  PHI
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(C6) elliptic_kc(1/2);
 
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(%i4) elliptic_e(phi,1);
 
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(%o4)                               SIN(PHI)
 
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(%i5) elliptic_e(phi,0);
 
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(%o5)                                  PHI
 
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(%i6) elliptic_kc(1/2);
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                                            1
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(D6)                            ELLIPTIC_KC(-)
 
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(%o6)                            ELLIPTIC_KC(-)
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                                            2
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(C7) makegamma(%);
 
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(%i7) makegamma(%);
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                                        2 1
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                                   GAMMA (-)
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                                          4
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(D7)                              -----------
 
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(%o7)                              -----------
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                                  4 SQRT(%PI)
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(C8) diff(elliptic_f(phi,m),phi);
 
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(%i8) diff(elliptic_f(phi,m),phi);
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                                       1
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(D8)                         ---------------------
 
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(%o8)                         ---------------------
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                                           2
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                             SQRT(1 - m SIN (PHI))
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(C9) diff(elliptic_f(phi,m),m);
 
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(%i9) diff(elliptic_f(phi,m),m);
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      ELLIPTIC_E(PHI, m) - (1 - m) ELLIPTIC_F(PHI, m)
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(D9) (-----------------------------------------------
 
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(%o9) (-----------------------------------------------
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                             m
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                            COS(PHI) SIN(PHI)