version 1.3, 2000/01/03 09:15:53 |
version 1.6, 2000/02/20 11:30:44 |
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@c $OpenXM$ |
@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. |
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@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$^$9(B. |
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$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 $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$ |
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Line 13 $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$ |
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$$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$ |
$$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$ |
$B$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$,$"$j$^$9(B. |
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$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$$B$GI=8=$5$l$^$9(B. |
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$BNc$($P(B$x \partial_{x} + y \partial_{y} + 1$$B$K$D$$$F$O(B |
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$$ x*dx+y*dy+1 $$ |
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$B$HI=8=$5$l$^$9(B. |
@end tex |
@end tex |
@menu |
@menu |
@c * diff_op_hg1:: |
@c * diff_op_hg1:: |
Line 26 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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Line 31 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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* diff_op_appell4:: |
* diff_op_appell4:: |
@c * diff_op_selberg2:: |
@c * diff_op_selberg2:: |
@c * diff_op_gkz:: |
@c * diff_op_gkz:: |
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* diff_op_tosm1:: |
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* diff_op_toasir:: |
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* diff_op_fromasir:: |
* diff_act:: |
* diff_act:: |
@c * diff_act_hg1:: |
@c * diff_act_hg1:: |
@c * diff_act_appell1:: |
@c * diff_act_appell1:: |
Line 35 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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Line 43 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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@c * diff_act_selberg2:: |
@c * diff_act_selberg2:: |
@c * diff_act_gkz:: |
@c * diff_act_gkz:: |
* diff_poly_solve:: |
* diff_poly_solve:: |
@c * diff_poly_solve_hg1:: |
* diff_poly_solve_hg1:: |
@c * diff_poly_solve_appell1:: |
@c * diff_poly_solve_appell1:: |
@c * diff_poly_solve_appell2:: |
@c * diff_poly_solve_appell2:: |
@c * diff_poly_solve_appell3:: |
@c * diff_poly_solve_appell3:: |
Line 72 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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Line 80 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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[(-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 |
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@node diff_op_tosm1,,, Differential equations |
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@subsection @code{diff_op_tosm1} |
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@findex diff_op_tosm1 |
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@table @t |
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@item diff_op_tosm1(@var{LL},@var{V}) |
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:: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$r(B sm1 $B7A<0$KJQ49$7$^$9(B. |
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@end table |
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@table @var |
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@item return |
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$B%j%9%H(B |
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@item LL |
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$B%j%9%H(B |
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@item V |
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$B%j%9%H(B |
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@end table |
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@itemize @bullet |
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@item $BHyJ,:nMQAG$N78?t$O@0?tB?9`<0$KJQ49$5$l$^$9(B. |
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@item @code{diff_op_tosm1}$B$NNc(B |
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@end itemize |
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@example |
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[299] diff_op_tosm1([[[x,[2,0]],[-1,[0,0]]], |
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[[y,[0,2]],[-1,[0,0]]]],[x,y]); |
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[ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))] |
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[300] diff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]], |
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[[1,[2,0]],[1,[0,2]]]],[x,y]); |
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[ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2] |
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[301] diff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]], |
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[[1/3,[0,1]],[1/4,[0,0]]]],[x,y]); |
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[ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))] |
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[302] diff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]], |
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[[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]); |
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[ + ( + (6) x) dx + ( + (12)), + ( + (4) y) dy + ( + (3))] |
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@end example |
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@node diff_op_toasir,,, Differential equations |
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@subsection @code{diff_op_toasir} |
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@findex diff_op_toasir |
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@table @t |
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@item diff_op_toasir(@var{LL},@var{V}) |
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:: $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. |
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@end table |
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@table @var |
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@item return |
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$B%j%9%H(B |
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@item LL |
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$B%j%9%H(B |
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@item V |
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$B%j%9%H(B |
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@end table |
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@itemize @bullet |
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@item @code{diff_op_toasir}$B$NNc(B |
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@end itemize |
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@example |
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[303] diff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]], |
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[[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]); |
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[1/2*x*dx+1,1/3*y*dy+1/4] |
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[304] diff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]], |
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[[1,[2,0]],[1,[0,2]]]],[x,y]); |
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[x*dx+y*dy+1,dx^2+dy^2] |
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@end example |
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@node diff_op_fromasir,,, Differential equations |
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@subsection @code{diff_op_fromasir} |
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@findex diff_op_fromasir |
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@table @t |
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@item diff_op_fromasir(@var{D_list},@var{V}) |
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:: @code{asir} $B$NB?9`<0$+$i%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$KJQ49$7$^$9(B. |
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@end table |
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@table @var |
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@item return |
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$B%j%9%H(B |
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@item D_list |
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$B%j%9%H(B |
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@item V |
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$B%j%9%H(B |
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@end table |
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@itemize @bullet |
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@item @code{diff_op_fromasir}$B$NNc(B |
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@end itemize |
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@example |
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[305] diff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]); |
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[[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]] |
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[306] diff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]); |
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[[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]] |
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@end example |
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@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, |
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Line 182 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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@item return |
@item return |
$BM-M}<0(B |
$BM-M}<0(B |
@item L |
@item L |
$B%j%9%H(B |
$B%j%9%H(B or $BB?9`<0(B |
@item F |
@item F |
$BM-M}<0(B |
$BM-M}<0(B |
@item V |
@item V |
Line 98 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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Line 197 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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[303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]); |
[303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]); |
2*x+2*y |
2*x+2*y |
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[349] diff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]); |
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2*x^2+2*y*x+2*y^2 |
@end example |
@end example |
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@node diff_act_appell4,,, Differential equations |
@node diff_act_appell4,,, Differential equations |
Line 157 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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Line 259 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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[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]] |
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[356] diff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]); |
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[-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]] |
@end example |
@end example |
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@node diff_poly_solve_hg1,,, Differential equations |
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@subsection @code{diff_poly_solve_hg1} |
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@findex diff_poly_solve_hg1 |
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@table @t |
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@item diff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V}) |
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:: $B%,%&%9$ND64v2?HyJ,J}Dx<0$NB?9`<02r$r5a$a$k(B. |
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@end table |
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@table @var |
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@item return |
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$B%j%9%H(B |
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@item a, b, c |
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$BM-M}<0(B |
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@item V |
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$B%j%9%H(B |
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@end table |
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@itemize @bullet |
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@item @code{diff_poly_solve_hg1}$B$NNc(B. |
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@end itemize |
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@example |
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[334] diff_poly_solve_hg1(-3,-6,-5,[x]); |
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[_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]] |
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[335] diff_poly_solve_hg1(-3,-6,-7,[x]); |
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[-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]] |
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@end example |
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@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 |
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$B@0?t(B |
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@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, |
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Line 309 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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@item @code{diff_poly_solve_appell4}$B$NNc(B. |
@item @code{diff_poly_solve_appell4}$B$NNc(B. |
@end itemize |
@end itemize |
@example |
@example |
[299] diff_poly_solve_appell4(-3,1,-1,-1,5,[x,y]); |
[299] diff_poly_solve_appell4(-3,1,-1,-1,[x,y]); |
[-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]] |
[-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]] |
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[300] diff_poly_solve_appell4(-3,1,1,-1,5,[x,y]); |
[300] diff_poly_solve_appell4(-3,1,1,-1,[x,y]); |
[-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]] |
[-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]] |
@end example |
@end example |
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Line 210 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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Line 339 $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1, |
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@item @code{diff_rat_solve}$B$NNc(B. |
@item @code{diff_rat_solve}$B$NNc(B. |
@end itemize |
@end itemize |
@example |
@example |
@end example |
[333] diff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]); |
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[(_8)/(x),[_8]] |
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[361] diff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]); |
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[(_180)/(-x+1),[_180]] |
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[350] D = diff_op_appell4(0,0,3,0,[x,y])$ |
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[351] diff_rat_solve(D,x^2,2,[x,y]); |
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[(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]] |
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@end example |