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version 1.4, 2000/01/21 12:54:39 version 1.6, 2000/02/20 11:30:44
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 @c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.3 2000/01/03 09:15:53 okutani Exp $  @c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.5 2000/01/31 11:01:00 okutani Exp $
 @node Differential equations,,, $B$=$NB>$NH!?t(B  @node Differential equations,,, $B$=$NB>$NH!?t(B
 @section Differential equations  @section Differential equations
 $B%U%!%$%k(B @file{gr}, @file{Matrix} $B$,I,MW$G$9(B.  $B%U%!%$%k(B @file{gr}, @file{Matrix} $B$,I,MW$G$9(B.
   
 @tex  @tex
 $B$3$N@a$G>R2p$5$l$k4X?t$G$OHyJ,:nMQAG$O<!$N$h$&$J%j%9%H$GI=8=$7$^$9(B.  $B$3$N@a$G>R2p$5$l$k4X?t$G$OHyJ,:nMQAG$O%j%9%H$^$?$OB?9`<0$GI=8=$5$l$^$9(B.
   $B%j%9%H$K$h$kI=8=$O<!$N$h$&$K$J$j$^$9(B.
 $$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$  $$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$
 $B$3$l$O(B  $B$3$l$O(B
 $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$  $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
Line 12  $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
Line 13  $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
 $$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$  $$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$
 $B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.  $B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.
 $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$  $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$
 $BNc$($PHyJ,:nMQAG(B$x dx + y dy + 1$$B$N>l9g$O(B  $BNc$($PHyJ,:nMQAG(B$x \partial_{x} + y \partial_{y} + 1$$B$N>l9g$O(B
 $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$  $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$
 $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x dx + y dy + 1, dx^2 + dy^2$$B$rI=$9$H(B  $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x \partial_{x} + y \partial_{y} + 1, {\partial_{x}}^{2} + {\partial_{y}}^{2}$$B$rI=$9$H(B
 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$
 $B$H$J$j$^$9(B. $B$^$?$3$l$i$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9!#(B  $B$H$J$j$^$9(B. $B$^$?$3$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9(B.
   $B<!$KB?9`<0$K$h$kI=8=$K$D$$$F=R$Y$^$9(B. $BJQ?t(B$x$$B$KBP$9$kHyJ,$O(B$dx$$B$GI=8=$5$l$^$9(B.
   $BNc$($P(B$x \partial_{x} + y \partial_{y} + 1$$B$K$D$$$F$O(B
   $$ x*dx+y*dy+1 $$
   $B$HI=8=$5$l$^$9(B.
 @end tex  @end tex
 @menu  @menu
 @c * diff_op_hg1::  @c * diff_op_hg1::
Line 26  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 31  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
 * diff_op_appell4::  * diff_op_appell4::
 @c * diff_op_selberg2::  @c * diff_op_selberg2::
 @c * diff_op_gkz::  @c * diff_op_gkz::
   * diff_op_tosm1::
   * diff_op_toasir::
   * diff_op_fromasir::
 * diff_act::  * diff_act::
 @c * diff_act_hg1::  @c * diff_act_hg1::
 @c * diff_act_appell1::  @c * diff_act_appell1::
Line 35  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 43  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
 @c * diff_act_selberg2::  @c * diff_act_selberg2::
 @c * diff_act_gkz::  @c * diff_act_gkz::
 * diff_poly_solve::  * diff_poly_solve::
 @c * diff_poly_solve_hg1::  * diff_poly_solve_hg1::
 @c * diff_poly_solve_appell1::  @c * diff_poly_solve_appell1::
 @c * diff_poly_solve_appell2::  @c * diff_poly_solve_appell2::
 @c * diff_poly_solve_appell3::  @c * diff_poly_solve_appell3::
Line 72  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 80  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
     [(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]      [(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]
 @end example  @end example
   
   @node diff_op_tosm1,,, Differential equations
   @subsection @code{diff_op_tosm1}
   @findex diff_op_tosm1
   @table @t
   @item diff_op_tosm1(@var{LL},@var{V})
   ::  $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$r(B sm1 $B7A<0$KJQ49$7$^$9(B.
   @end table
   @table @var
   @item return
   $B%j%9%H(B
   @item LL
   $B%j%9%H(B
   @item V
   $B%j%9%H(B
   @end table
   @itemize @bullet
   @item  $BHyJ,:nMQAG$N78?t$O@0?tB?9`<0$KJQ49$5$l$^$9(B.
   @item  @code{diff_op_tosm1}$B$NNc(B
   @end itemize
   @example
   [299] diff_op_tosm1([[[x,[2,0]],[-1,[0,0]]],
                             [[y,[0,2]],[-1,[0,0]]]],[x,y]);
   [ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))]
   
   [300] diff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
                             [[1,[2,0]],[1,[0,2]]]],[x,y]);
   [ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2]
   
   [301] diff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]],
                             [[1/3,[0,1]],[1/4,[0,0]]]],[x,y]);
   [ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))]
   
   [302] diff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]],
                             [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
   [ + ( + (6) x) dx + ( + (12)), + ( + (4) y) dy + ( + (3))]
   @end example
   
   @node diff_op_toasir,,, Differential equations
   @subsection @code{diff_op_toasir}
   @findex diff_op_toasir
   @table @t
   @item diff_op_toasir(@var{LL},@var{V})
   ::  $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H(B @var{LL} $B$r(B @code{asir} $B$NB?9`<0$KJQ49$7$^$9(B.
   @end table
   @table @var
   @item return
   $B%j%9%H(B
   @item LL
   $B%j%9%H(B
   @item V
   $B%j%9%H(B
   @end table
   @itemize @bullet
   @item  @code{diff_op_toasir}$B$NNc(B
   @end itemize
   @example
   [303] diff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]],
                              [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
   [1/2*x*dx+1,1/3*y*dy+1/4]
   
   [304] diff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
                              [[1,[2,0]],[1,[0,2]]]],[x,y]);
   [x*dx+y*dy+1,dx^2+dy^2]
   @end example
   
   @node diff_op_fromasir,,, Differential equations
   @subsection @code{diff_op_fromasir}
   @findex diff_op_fromasir
   @table @t
   @item diff_op_fromasir(@var{D_list},@var{V})
   ::  @code{asir} $B$NB?9`<0$+$i%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$KJQ49$7$^$9(B.
   @end table
   @table @var
   @item return
   $B%j%9%H(B
   @item D_list
   $B%j%9%H(B
   @item V
   $B%j%9%H(B
   @end table
   @itemize @bullet
   @item  @code{diff_op_fromasir}$B$NNc(B
   @end itemize
   @example
   [305] diff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]);
   [[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]]
   
   [306] diff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]);
   [[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]]
   @end example
   
 @node diff_act,,, Differential equations  @node diff_act,,, Differential equations
 @subsection @code{diff_act}  @subsection @code{diff_act}
 @findex diff_act  @findex diff_act
Line 83  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 182  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
 @item return  @item return
 $BM-M}<0(B  $BM-M}<0(B
 @item L  @item L
 $B%j%9%H(B  $B%j%9%H(B or $BB?9`<0(B
 @item F  @item F
 $BM-M}<0(B  $BM-M}<0(B
 @item V  @item V
Line 98  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 197  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
   
 [303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);  [303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);
 2*x+2*y  2*x+2*y
   
   [349] diff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]);
   2*x^2+2*y*x+2*y^2
 @end example  @end example
   
 @node diff_act_appell4,,, Differential equations  @node diff_act_appell4,,, Differential equations
Line 157  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 259  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
   
 [298] diff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);  [298] diff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);
 [_33*y^2*x^2,[_33]]  [_33*y^2*x^2,[_33]]
   
   [356] diff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]);
   [-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]]
 @end example  @end example
   
   @node diff_poly_solve_hg1,,, Differential equations
   @subsection @code{diff_poly_solve_hg1}
   @findex diff_poly_solve_hg1
   @table @t
   @item diff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})
   ::  $B%,%&%9$ND64v2?HyJ,J}Dx<0$NB?9`<02r$r5a$a$k(B.
   @end table
   @table @var
   @item return
   $B%j%9%H(B
   @item a, b, c
   $BM-M}<0(B
   @item V
   $B%j%9%H(B
   @end table
   @itemize @bullet
   @item  @code{diff_poly_solve_hg1}$B$NNc(B.
   @end itemize
   @example
   [334] diff_poly_solve_hg1(-3,-6,-5,[x]);
   [_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]]
   
   [335] diff_poly_solve_hg1(-3,-6,-7,[x]);
   [-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]]
   @end example
   
 @node diff_poly_solve_appell4,,, Differential equations  @node diff_poly_solve_appell4,,, Differential equations
 @subsection @code{diff_poly_solve_appell4}  @subsection @code{diff_poly_solve_appell4}
 @findex diff_poly_solve_appell4  @findex diff_poly_solve_appell4
 @table @t  @table @t
 @item diff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{N},@var{V})  @item diff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
 ::  F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.  ::  F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$NB?9`<02r$r5a$a$k(B.
 @end table  @end table
 @table @var  @table @var
 @item return  @item return
 $B%j%9%H(B  $B%j%9%H(B
 @item a, b, c1, c2  @item a, b, c1, c2
 $BM-M}<0(B  $BM-M}<0(B
 @item N  
 $B@0?t(B  
 @item V  @item V
 $B%j%9%H(B  $B%j%9%H(B
 @end table  @end table
Line 180  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 309  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
 @item  @code{diff_poly_solve_appell4}$B$NNc(B.  @item  @code{diff_poly_solve_appell4}$B$NNc(B.
 @end itemize  @end itemize
 @example  @example
 [299] diff_poly_solve_appell4(-3,1,-1,-1,5,[x,y]);  [299] diff_poly_solve_appell4(-3,1,-1,-1,[x,y]);
 [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]  [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]
   
 [300] diff_poly_solve_appell4(-3,1,1,-1,5,[x,y]);  [300] diff_poly_solve_appell4(-3,1,1,-1,[x,y]);
 [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]  [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]
 @end example  @end example
   
Line 212  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
Line 341  $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,
 @example  @example
 [333] diff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);  [333] diff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);
 [(_8)/(x),[_8]]  [(_8)/(x),[_8]]
   
   [361] diff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]);
   [(_180)/(-x+1),[_180]]
   
 [350] D = diff_op_appell4(0,0,3,0,[x,y])$  [350] D = diff_op_appell4(0,0,3,0,[x,y])$
 [351] diff_rat_solve(D,x^2,2,[x,y]);  [351] diff_rat_solve(D,x^2,2,[x,y]);
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