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apply1

Run Example
(%i1)remvalue(n,z);
(%o1)                           [false, false]
(%i2) matchdeclare(a,lambda([x],freeof(x,n)));
(%o2)                                done
(%i3) matchdeclare(b,lambda([x],freeof(x,n) and integerp(x) and x>
0));
(%o3)                                done
(%i4) matchdeclare(f,all);
(%o4)                                done
(%i5) matchdeclare(g,lambda([x],freeof(n-1,x)));
(%o5)                                done
(%i6) defrule(r1,zt(1,n,z),z/(z-1));
                                                 z
(%o6)                      r1 : zt(1, n, z) -> -----
                                               z - 1
(%i7) defrule(r2,zt(n,n,z),z/(z-1)^2);
                                                z
(%o7)                    r2 : zt(n, n, z) -> --------
                                                    2
                                             (z - 1)
(%i8) defrule(r3,zt(1/n,n,z),log(z/(z-1)));
                                1                 z
(%o8)                   r3 : zt(-, n, z) -> log(-----)
                                n               z - 1
(%i9) defrule(r4,zt(a^n,n,z),z/(z-a));
                                   n             z
(%o9)                     r4 : zt(a , n, z) -> -----
                                               z - a
(%i10) defrule(r5,zt(a^(n-1),n,z),1/(z-a));
                                 n - 1             1
(%o10)                  r5 : zt(a     , n, z) -> -----
                                                 z - a
(%i11) defrule(r6,zt(binomial(n,b),n,z),z/(z-1)^(b+1));
                                                       - b - 1
(%o11)         r6 : zt(binomial(n, b), n, z) -> (z - 1)        z
(%i12) defrule(r7,zt(n^b,n,z),block([tmp:z/(z-1)],  for i:1 thru b do tmp:-z*diff(tmp,z),  factor(tmp)));
                b                          z
(%o12) r7 : zt(n , n, z) -> block([tmp : -----], 
                                         z - 1
                           for i thru b do tmp : - diff(tmp, z) z, factor(tmp))
(%i13) defrule(r8,zt(sin(a*n),n,z),z*'sin(a)/((z^2-2*z*'cos(a)+1)));
                                                sin(a) z
(%o13)          r8 : zt(sin(a n), n, z) -> -------------------
                                            2
                                           z  - 2 cos(a) z + 1
(%i14) defrule(r9,zt(cos(a*n),n,z),z*(z-'cos(a))/(z^2-2*z*'cos(a)+1));
                                             z (z - cos(a))
(%o14)          r9 : zt(cos(a n), n, z) -> -------------------
                                            2
                                           z  - 2 cos(a) z + 1
(%i15) defrule(r10,zt(kron_delta(b,n),n,z),1/z^b);
                                                        1
(%o15)              r10 : zt(kron_delta(b, n), n, z) -> --
                                                         b
                                                        z
(%i16) defrule(r11,zt(a^n*f,n,z),'subst(z/a,z,ztrans(f,n,z)));
                     n                        z
(%o16)     r11 : zt(a  f, n, z) -> substitute(-, z, ztrans(f, n, z))
                                              a
(%i17) defrule(r12,zt(n^b*g,n,z),block([tmp:ztrans(g,n,z)],  for i:1 thru b do tmp:-z*'diff(tmp,z),  return(tmp)));
                   b
(%o17) r12 : zt(g n , n, z) -> block([tmp : ztrans(g, n, z)], 
                                                           dtmp
                                   for i thru b do tmp : - ---- z, return(tmp))
                                                            dz
(%i18) defrule(r13,zt(exp(-a*n),n,z),z/(z-exp(-a)));
                                - a n               z
(%o18)               r13 : zt(%e     , n, z) -> ---------
                                                      - a
                                                z - %e
(%i19) defrule(r14,zt(f(n-m),n,z),z^(-m)*ztrans(f(n),n,z));
                                            ztrans(f(n), n, z)
(%o19)          r14 : zt(f(n - m), n, z) -> ------------------
                                                     m
                                                    z
(%i20) declare(zt,linear);
(%o20)                               done
(%i21) ztrans1(e):=apply1(e,r1,r2,r3,r4,r5,r6,r7,r8,r9,r10,r11,r12,r13,r14);
(%o21) ztrans1(e) := apply1(e, r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, 
                                                                 r12, r13, r14)
(%i22) ztrans(f,n,z):=ev(ztrans1(zt(f,n,z)),diff,subst);
(%o22)     ztrans(f, n, z) := ev(ztrans1(zt(f, n, z)), diff, subst)
(%i23) defrule(ir1,izt(z/(z-a),z,n),a^n);
                                     z              n
(%o23)                   ir1 : izt(-----, z, n) -> a
                                   z - a
(%i24) defrule(ir2,izt(1/(z-a),z,n),a^(n-1));
                                   1              n - 1
(%o24)                 ir2 : izt(-----, z, n) -> a
                                 z - a
(%i25) declare(izt,linear);
(%o25)                               done
(%i26) invztrans1(e):=apply1(e,ir1,ir2);
(%o26)               invztrans1(e) := apply1(e, ir1, ir2)
(%i27) invztrans(f,z,n):=invztrans1(izt(f,z,n));
(%o27)          invztrans(f, z, n) := invztrans1(izt(f, z, n))
(%i28) ztrans(1,n,z);
                                       z
(%o28)                               -----
                                     z - 1
(%i29) ztrans(a,n,z);
                                      a z
(%o29)                               -----
                                     z - 1
(%i30) expand(ztrans((-1)^n,n,z));
                                       z
(%o30)                               -----
                                     z + 1
(%i31) ztrans(n,n,z);
                                      z
(%o31)                             --------
                                          2
                                   (z - 1)
(%i32) ztrans(n^2,n,z);
                                   z (z + 1)
(%o32)                             ---------
                                          3
                                   (z - 1)
(%i33) ztrans(a^(n-1),n,z);
                                       1
(%o33)                               -----
                                     z - a
(%i34) factor(ztrans(a^(n-1)+n^2,n,z));
                         3      2      2
                      2 z  - a z  - 2 z  - a z + 3 z - 1
(%o34)                ----------------------------------
                                      3
                               (z - 1)  (z - a)
(%i35) 
Run Example
load(vect);
(%o1)           /usr/share/maxima/5.21.1/share/vector/vect.mac
(%i2) load(eigen);
(%o2)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i3) load(linearalgebra);
(%o3)   /usr/share/maxima/5.21.1/share/linearalgebra/linearalgebra.mac
(%i4) assume(a>
0,b>
0,c>
0);
(%o4)                        [a > 0, b > 0, c > 0]
(%i5) g(x,y,z,t):=((x^2)/(a^2+t)+(y^2)/(b^2+t)+(z^2)/(c^2+t)-1);
                                     2        2        2
                                    x        y        z
(%o5)            g(x, y, z, t) := ------ + ------ + ------ - 1
                                   2        2        2
                                  a  + t   b  + t   c  + t
(%i6) f(t):=sqrt((a^2+t)*(b^2+t)*(c^2+t));
                                  2        2        2
(%o6)              f(t) := sqrt((a  + t) (b  + t) (c  + t))
(%i7) G(n,l):='integrate((g(x,y,z,t)^n)/f(t),t,l,inf);
                                   inf
                                  /     n
                                  [    g (x, y, z, t)
(%o7)                  G(n, l) := I    -------------- dt
                                  ]         f(t)
                                  /
                                   l
(%i8) G1:G(1,l(x,y,z));
                                 2        2        2
                                z        y        x
                 inf          ------ + ------ + ------ - 1
                /                  2        2        2
                [             t + c    t + b    t + a
(%o8)           I           -------------------------------- dt
                ]                      2        2        2
                /           sqrt((t + a ) (t + b ) (t + c ))
                 l(x, y, z)
(%i9) G2:G(2,l(x,y,z));
                                2        2        2
                               z        y        x        2
                 inf        (------ + ------ + ------ - 1)
                /                 2        2        2
                [            t + c    t + b    t + a
(%o9)           I           -------------------------------- dt
                ]                      2        2        2
                /           sqrt((t + a ) (t + b ) (t + c ))
                 l(x, y, z)
(%i10) kron:ident(3);
                                  [ 1  0  0 ]
                                  [         ]
(%o10)                            [ 0  1  0 ]
                                  [         ]
                                  [ 0  0  1 ]
(%i11) aye:matrix([a^2,b^2,c^2]/4);
                                [  2   2   2 ]
(%o11)                          [ a   b   c  ]
                                [ --  --  -- ]
                                [ 4   4   4  ]
(%i12) eks:matrix([x,y,z]);
(%o12)                            [ x  y  z ]
(%i13) defrule(g0,g(x,y,z,l(x,y,z)),0);
(%o13)                 g0 : g(x, y, z, l(x, y, z)) -> 0
(%i14) one:kron*G1;
                [                  2        2        2            ]
                [                 z        y        x             ]
                [  inf          ------ + ------ + ------ - 1      ]
                [ /                  2        2        2          ]
                [ [             t + c    t + b    t + a           ]
                [ I           -------------------------------- dt ]
(%o14)  Col 1 = [ ]                      2        2        2      ]
                [ /           sqrt((t + a ) (t + b ) (t + c ))    ]
                [  l(x, y, z)                                     ]
                [                                                 ]
                [                        0                        ]
                [                                                 ]
                [                        0                        ]
         [                        0                        ]
         [                                                 ]
         [                  2        2        2            ]
         [                 z        y        x             ]
         [  inf          ------ + ------ + ------ - 1      ]
         [ /                  2        2        2          ]
 Col 2 = [ [             t + c    t + b    t + a           ]
         [ I           -------------------------------- dt ]
         [ ]                      2        2        2      ]
         [ /           sqrt((t + a ) (t + b ) (t + c ))    ]
         [  l(x, y, z)                                     ]
         [                                                 ]
         [                        0                        ]
         [                        0                        ]
         [                                                 ]
         [                        0                        ]
         [                                                 ]
         [                  2        2        2            ]
         [                 z        y        x             ]
 Col 3 = [  inf          ------ + ------ + ------ - 1      ]
         [ /                  2        2        2          ]
         [ [             t + c    t + b    t + a           ]
         [ I           -------------------------------- dt ]
         [ ]                      2        2        2      ]
         [ /           sqrt((t + a ) (t + b ) (t + c ))    ]
         [  l(x, y, z)                                     ]
(%i15) two:transpose(-eks).matrix(apply1(ev(express(grad(G1)),diff),g0));
                     inf
                    /
                    [                               1
(%o15) matrix([- 2 (I           ----------------------------------------- dt)
                    ]                 2             2        2        2
                    /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
                     l(x, y, z)
           inf
          /
  2       [                               1
 x , - 2 (I           ----------------------------------------- dt) x y, 
          ]                 2             2        2        2
          /           (t + b ) sqrt((t + a ) (t + b ) (t + c ))
           l(x, y, z)
      inf
     /
     [                               1
- 2 (I           ----------------------------------------- dt) x z], 
     ]                 2             2        2        2
     /           (t + c ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
       inf
      /
      [                               1
[- 2 (I           ----------------------------------------- dt) x y, 
      ]                 2             2        2        2
      /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
       l(x, y, z)
      inf
     /
     [                               1                          2
- 2 (I           ----------------------------------------- dt) y , 
     ]                 2             2        2        2
     /           (t + b ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
      inf
     /
     [                               1
- 2 (I           ----------------------------------------- dt) y z], 
     ]                 2             2        2        2
     /           (t + c ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
       inf
      /
      [                               1
[- 2 (I           ----------------------------------------- dt) x z, 
      ]                 2             2        2        2
      /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
       l(x, y, z)
      inf
     /
     [                               1
- 2 (I           ----------------------------------------- dt) y z, 
     ]                 2             2        2        2
     /           (t + b ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
      inf
     /
     [                               1                          2
- 2 (I           ----------------------------------------- dt) z ])
     ]                 2             2        2        2
     /           (t + c ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
(%i16) two[1][1];
             inf
            /
            [                               1                          2
(%o16) - 2 (I           ----------------------------------------- dt) x
            ]                 2             2        2        2
            /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
             l(x, y, z)
(%i17) 
Run Example
load(vect);
(%o1)           /usr/share/maxima/5.21.1/share/vector/vect.mac
(%i2) load(eigen);
(%o2)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%i3) load(linearalgebra);
(%o3)   /usr/share/maxima/5.21.1/share/linearalgebra/linearalgebra.mac
(%i4) assume(a>
0,b>
0,c>
0);
(%o4)                        [a > 0, b > 0, c > 0]
(%i5) g(x,y,z,t):=((x^2)/(a^2+t)+(y^2)/(b^2+t)+(z^2)/(c^2+t)-1);
                                     2        2        2
                                    x        y        z
(%o5)            g(x, y, z, t) := ------ + ------ + ------ - 1
                                   2        2        2
                                  a  + t   b  + t   c  + t
(%i6) f(t):=sqrt((a^2+t)*(b^2+t)*(c^2+t));
                                  2        2        2
(%o6)              f(t) := sqrt((a  + t) (b  + t) (c  + t))
(%i7) G(n,l):='integrate((g(x,y,z,t)^n)/f(t),t,l,inf);
                                   inf
                                  /     n
                                  [    g (x, y, z, t)
(%o7)                  G(n, l) := I    -------------- dt
                                  ]         f(t)
                                  /
                                   l
(%i8) G1:G(1,l(x,y,z));
                                 2        2        2
                                z        y        x
                 inf          ------ + ------ + ------ - 1
                /                  2        2        2
                [             t + c    t + b    t + a
(%o8)           I           -------------------------------- dt
                ]                      2        2        2
                /           sqrt((t + a ) (t + b ) (t + c ))
                 l(x, y, z)
(%i9) G2:G(2,l(x,y,z));
                                2        2        2
                               z        y        x        2
                 inf        (------ + ------ + ------ - 1)
                /                 2        2        2
                [            t + c    t + b    t + a
(%o9)           I           -------------------------------- dt
                ]                      2        2        2
                /           sqrt((t + a ) (t + b ) (t + c ))
                 l(x, y, z)
(%i10) kron:ident(3);
                                  [ 1  0  0 ]
                                  [         ]
(%o10)                            [ 0  1  0 ]
                                  [         ]
                                  [ 0  0  1 ]
(%i11) aye:[a^2,b^2,c^2]/4;
                                   2   2   2
                                  a   b   c
(%o11)                           [--, --, --]
                                  4   4   4
(%i12) eks:matrix([x,y,z]);
(%o12)                            [ x  y  z ]
(%i13) defrule(g0,g(x,y,z,l(x,y,z)),0);
(%o13)                 g0 : g(x, y, z, l(x, y, z)) -> 0
(%i14) two:transpose(-eks).matrix(apply1(ev(express(grad(G1)),diff),g0));
                     inf
                    /
                    [                               1
(%o14) matrix([- 2 (I           ----------------------------------------- dt)
                    ]                 2             2        2        2
                    /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
                     l(x, y, z)
           inf
          /
  2       [                               1
 x , - 2 (I           ----------------------------------------- dt) x y, 
          ]                 2             2        2        2
          /           (t + b ) sqrt((t + a ) (t + b ) (t + c ))
           l(x, y, z)
      inf
     /
     [                               1
- 2 (I           ----------------------------------------- dt) x z], 
     ]                 2             2        2        2
     /           (t + c ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
       inf
      /
      [                               1
[- 2 (I           ----------------------------------------- dt) x y, 
      ]                 2             2        2        2
      /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
       l(x, y, z)
      inf
     /
     [                               1                          2
- 2 (I           ----------------------------------------- dt) y , 
     ]                 2             2        2        2
     /           (t + b ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
      inf
     /
     [                               1
- 2 (I           ----------------------------------------- dt) y z], 
     ]                 2             2        2        2
     /           (t + c ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
       inf
      /
      [                               1
[- 2 (I           ----------------------------------------- dt) x z, 
      ]                 2             2        2        2
      /           (t + a ) sqrt((t + a ) (t + b ) (t + c ))
       l(x, y, z)
      inf
     /
     [                               1
- 2 (I           ----------------------------------------- dt) y z, 
     ]                 2             2        2        2
     /           (t + b ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
      inf
     /
     [                               1                          2
- 2 (I           ----------------------------------------- dt) z ])
     ]                 2             2        2        2
     /           (t + c ) sqrt((t + a ) (t + b ) (t + c ))
      l(x, y, z)
(%i15) defrule(X,diff(l(x,y,z),x),XXX);
(%o15)                  X : diff(l(x, y, z), x) -> XXX
(%i16) defrule(Y,diff(l(x,y,z),y),YYY);
(%o16)                  Y : diff(l(x, y, z), y) -> YYY
(%i17) defrule(Z,diff(l(x,y,z),z),ZZZ);
(%o17)                  Z : diff(l(x, y, z), z) -> ZZZ
(%i18) defrule(L,l(x,y,z),0);
(%o18)                        L : l(x, y, z) -> 0
(%i19) two_test:apply1(ev(express(grad(two)),diff),g0,X,Y,Z,L);
                   2
                2 x  XXX
(%o19) [matrix([-------- - 4
                  3
                 a  b c
   inf
  /
  [                        1
 (I    ----------------------------------------- dt) x, 
  ]          2             2        2        2
  /    (t + a ) sqrt((t + a ) (t + b ) (t + c ))
   0
                inf
               /
2 x y XXX      [                        1
--------- - 2 (I    ----------------------------------------- dt) y, 
    3          ]          2             2        2        2
 a b  c        /    (t + b ) sqrt((t + a ) (t + b ) (t + c ))
                0
                inf
               /
2 x z XXX      [                        1
--------- - 2 (I    ----------------------------------------- dt) z], 
      3        ]          2             2        2        2
 a b c         /    (t + c ) sqrt((t + a ) (t + b ) (t + c ))
                0
                 inf
                /
 2 x y XXX      [                        1
[--------- - 2 (I    ----------------------------------------- dt) y, 
   3            ]          2             2        2        2
  a  b c        /    (t + a ) sqrt((t + a ) (t + b ) (t + c ))
                 0
   2
2 y  XXX  2 y z XXX    2 x z XXX
--------, ---------], [--------- - 2
    3           3        3
 a b  c    a b c        a  b c
   inf
  /                                                                   2
  [                        1                            2 y z XXX  2 z  XXX
 (I    ----------------------------------------- dt) z, ---------, --------]), 
  ]          2             2        2        2              3            3
  /    (t + a ) sqrt((t + a ) (t + b ) (t + c ))         a b  c     a b c
   0
           2
        2 x  YYY  2 x y YYY
matrix([--------, --------- - 2
          3           3
         a  b c    a b  c
   inf
  /
  [                        1                            2 x z YYY
 (I    ----------------------------------------- dt) x, ---------], 
  ]          2             2        2        2                3
  /    (t + b ) sqrt((t + a ) (t + b ) (t + c ))         a b c
   0
                 inf
                /
 2 x y YYY      [                        1
[--------- - 2 (I    ----------------------------------------- dt) x, 
   3            ]          2             2        2        2
  a  b c        /    (t + a ) sqrt((t + a ) (t + b ) (t + c ))
                 0
               inf
   2          /
2 y  YYY      [                        1
-------- - 4 (I    ----------------------------------------- dt) y, 
    3         ]          2             2        2        2
 a b  c       /    (t + b ) sqrt((t + a ) (t + b ) (t + c ))
               0
                inf
               /
2 y z YYY      [                        1
--------- - 2 (I    ----------------------------------------- dt) z], 
      3        ]          2             2        2        2
 a b c         /    (t + c ) sqrt((t + a ) (t + b ) (t + c ))
                0
                            inf
                           /
 2 x z YYY  2 y z YYY      [                        1
[---------, --------- - 2 (I    ----------------------------------------- dt)
   3            3          ]          2             2        2        2
  a  b c     a b  c        /    (t + b ) sqrt((t + a ) (t + b ) (t + c ))
                            0
       2                   2
    2 z  YYY            2 x  ZZZ  2 x y ZZZ
 z, --------]), matrix([--------, ---------, 
          3               3           3
     a b c               a  b c    a b  c
                inf
               /
2 x z ZZZ      [                        1
--------- - 2 (I    ----------------------------------------- dt) x], 
      3        ]          2             2        2        2
 a b c         /    (t + c ) sqrt((t + a ) (t + b ) (t + c ))
                0
               2
 2 x y ZZZ  2 y  ZZZ  2 y z ZZZ
[---------, --------, --------- - 2
   3            3           3
  a  b c     a b  c    a b c
   inf
  /
  [                        1
 (I    ----------------------------------------- dt) y], 
  ]          2             2        2        2
  /    (t + c ) sqrt((t + a ) (t + b ) (t + c ))
   0
                 inf
                /
 2 x z ZZZ      [                        1
[--------- - 2 (I    ----------------------------------------- dt) x, 
   3            ]          2             2        2        2
  a  b c        /    (t + a ) sqrt((t + a ) (t + b ) (t + c ))
                 0
                inf
               /
2 y z ZZZ      [                        1
--------- - 2 (I    ----------------------------------------- dt) y, 
    3          ]          2             2        2        2
 a b  c        /    (t + b ) sqrt((t + a ) (t + b ) (t + c ))
                0
               inf
   2          /
2 z  ZZZ      [                        1
-------- - 4 (I    ----------------------------------------- dt) z])]
      3       ]          2             2        2        2
 a b c        /    (t + c ) sqrt((t + a ) (t + b ) (t + c ))
               0
(%i20) 

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