Related

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termp (e) := symbolp(...

matchdeclare (pterm, ...

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apply1-binomial-block-cos-defrule-delta-diff-ev-exp-factor-freeof-kron_delta-lambda-linear-log-matchdeclare-remvalue-return-sin-subst

remvalue(n,z);

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apply1-defrule-matchdeclare-true

matchdeclare(n, integ...

defrule(fnx, f(n*x), ...

apply1(f(3*y), fnx);

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apply1-binomial-block-cos-defrule-delta-diff-ev-exp-expand-factor-freeof-kron_delta-lambda-linear-log-matchdeclare-remvalue-return-sin-subst

remvalue(n,z);

matchdeclare(a,lambda...

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apply1-cos-defrule-sin-true

mathdeclare([x,y], tr...

defrule(extrig1,sin(x...

apply1(E,extrig1);

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apply1-binomial-block-cos-defrule-delta-diff-ev-exp-factor-freeof-kron_delta-lambda-linear-log-matchdeclare-ratsimp-remvalue-return-sin-subst

remvalue(n,z);

matchdeclare(a,lambda...

matchdeclare(b,lambda...

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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
(%o2)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%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)                                     ]
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
(%o2)           /usr/share/maxima/5.21.1/share/matrix/eigen.mac
(%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
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
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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