Annotation of OpenXM/src/asir-contrib/packages/doc/Diff.texi, Revision 1.4
1.4 ! okutani 1: @c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.3 2000/01/03 09:15:53 okutani Exp $
1.1 takayama 2: @node Differential equations,,, $B$=$NB>$NH!?t(B
3: @section Differential equations
4: $B%U%!%$%k(B @file{gr}, @file{Matrix} $B$,I,MW$G$9(B.
5:
6: @tex
7: $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.
8: $$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$
9: $B$3$l$O(B
10: $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
11: $B$H$$$&0UL#$G$9(B. $B@~7?JPHyJ,J}Dx<07O(B
12: $$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$
13: $B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.
14: $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$
15: $BNc$($PHyJ,:nMQAG(B$x dx + y dy + 1$$B$N>l9g$O(B
16: $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$
17: $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x dx + y dy + 1, dx^2 + dy^2$$B$rI=$9$H(B
18: $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$
1.2 okutani 19: $B$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
1.1 takayama 20: @end tex
21: @menu
22: @c * diff_op_hg1::
23: @c * diff_op_appell1::
24: @c * diff_op_appell2::
25: @c * diff_op_appell3::
26: * diff_op_appell4::
27: @c * diff_op_selberg2::
28: @c * diff_op_gkz::
29: * diff_act::
30: @c * diff_act_hg1::
31: @c * diff_act_appell1::
32: @c * diff_act_appell2::
33: @c * diff_act_appell3::
34: * diff_act_appell4::
35: @c * diff_act_selberg2::
36: @c * diff_act_gkz::
37: * diff_poly_solve::
38: @c * diff_poly_solve_hg1::
39: @c * diff_poly_solve_appell1::
40: @c * diff_poly_solve_appell2::
41: @c * diff_poly_solve_appell3::
1.2 okutani 42: * diff_poly_solve_appell4::
1.1 takayama 43: @c * diff_poly_solve_selberg2::
44: @c * diff_poly_solve_gkz::
1.3 okutani 45: * diff_rat_solve::
1.2 okutani 46: @c * diff_pseries_appell4::
1.1 takayama 47: @end menu
48:
49: @node diff_op_appell4,,, Differential equations
50: @subsection @code{diff_op_appell4}
51: @findex diff_op_appell4
52: @table @t
53: @item diff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
54: :: appell $B$N(B F_4 $B$rNm2=$9$kHyJ,:nMQAG$r@8@.$7$^$9(B.
55: @end table
56: @table @var
57: @item return
58: $B%j%9%H(B
59: @item a, b, c1, c2
60: $BM-M}<0(B
61: @item V
62: $B%j%9%H(B
63: @end table
64: @itemize @bullet
65: @item @code{diff_op_appell4}$B$NNc(B.
66: @end itemize
67: @example
68: [298] diff_op_appell4(a,b,c1,c2,[x,y]);
69: [ [ [-x^2+x,[2,0]], [-2*y*x,[1,1]], [-y^2,[0,2]],
70: [(-a-b-1)*x+c1,[1,0]], [(-a-b-1)*y,[0,1]], [-b*a,[0,0]] ],
71: [ [-y^2+y,[0,2]], [-2*y*x,[1,1]], [-x^2,[2,0]],
72: [(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]
73: @end example
74:
75: @node diff_act,,, Differential equations
76: @subsection @code{diff_act}
77: @findex diff_act
78: @table @t
79: @item diff_act(@var{L},@var{F},@var{V})
80: :: $BHyJ,:nMQAG(B @var{L} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B. @var{V} $B$OJQ?t%j%9%H(B.
81: @end table
82: @table @var
83: @item return
84: $BM-M}<0(B
85: @item L
86: $B%j%9%H(B
87: @item F
88: $BM-M}<0(B
89: @item V
90: $B%j%9%H(B
91: @end table
92: @itemize @bullet
93: @item @code{diff_act}$B$NNc(B
94: @end itemize
95: @example
96: [302] diff_act([[1,[2]]],x^3+x^2+x+1,[x]);
97: 6*x+2
98:
99: [303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);
100: 2*x+2*y
101: @end example
102:
103: @node diff_act_appell4,,, Differential equations
104: @subsection @code{diff_act_appell4}
105: @findex diff_act_appell4
106: @table @t
107: @item diff_act_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{F},@var{V})
108: :: $BHyJ,:nMQAG(B @code{diff_op_appell4} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B.
109: @end table
110: @table @var
111: @item return
112: $B%j%9%H(B
113: @item a, b, c1, c2
114: $BM-M}<0(B
115: @item F
116: $BM-M}<0(B
117: @item V
118: $B%j%9%H(B
119: @end table
120: @itemize @bullet
121: @item @code{diff_act_appell4}$B$NNc(B
122: @end itemize
123: @example
124: [303] diff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);
125: [-6*x^2+4*x-6*y^2,-6*x^2-6*y^2+4*y]
126:
127: [304] diff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);
128: [-4*x^2+4*x-4*y^2,-4*x^2-4*y^2+4*y]
129:
130: [305] diff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);
131: [0,0]
132: @end example
133:
134: @node diff_poly_solve,,, Differential equations
135: @subsection @code{diff_poly_solve}
136: @findex diff_poly_solve
137: @table @t
138: @item diff_poly_solve(@var{LL},@var{N},@var{V})
139: :: $BM?$($i$l$?@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.
140: @end table
141: @table @var
142: @item return
143: $B%j%9%H(B
144: @item LL
145: $B%j%9%H(B
146: @item N
147: $B@0?t(B
148: @item V
149: $B%j%9%H(B
150: @end table
151: @itemize @bullet
152: @item @code{diff_poly_solve}$B$NNc(B.
153: @end itemize
154: @example
155: [297] diff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);
156: [_4*y*x,[_4]]
157:
158: [298] diff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);
159: [_33*y^2*x^2,[_33]]
1.2 okutani 160: @end example
161:
162: @node diff_poly_solve_appell4,,, Differential equations
163: @subsection @code{diff_poly_solve_appell4}
164: @findex diff_poly_solve_appell4
165: @table @t
166: @item diff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{N},@var{V})
167: :: F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.
168: @end table
169: @table @var
170: @item return
171: $B%j%9%H(B
172: @item a, b, c1, c2
173: $BM-M}<0(B
174: @item N
175: $B@0?t(B
176: @item V
177: $B%j%9%H(B
178: @end table
179: @itemize @bullet
180: @item @code{diff_poly_solve_appell4}$B$NNc(B.
181: @end itemize
182: @example
183: [299] diff_poly_solve_appell4(-3,1,-1,-1,5,[x,y]);
184: [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]
185:
186: [300] diff_poly_solve_appell4(-3,1,1,-1,5,[x,y]);
187: [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]
1.3 okutani 188: @end example
189:
190: @node diff_rat_solve,,, Differential equations
191: @subsection @code{diff_rat_solve}
192: @findex diff_rat_solve
193: @table @t
194: @item diff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})
195: :: $BM?$($i$l$?@~7?HyJ,J}Dx<07O$NJ,Jl$,(B @var{Dn}, $BJ,;R$,(B @var{N} $B<!0J2<$NB?9`<0$G$"$k$h$&$J2r$r5a$a$k(B.
196: @end table
197: @table @var
198: @item return
199: $B%j%9%H(B
200: @item LL
201: $B%j%9%H(B
202: @item Dn
203: $BM-M}<0(B
204: @item N
205: $B@0?t(B
206: @item V
207: $B%j%9%H(B
208: @end table
209: @itemize @bullet
210: @item @code{diff_rat_solve}$B$NNc(B.
211: @end itemize
212: @example
1.4 ! okutani 213: [333] diff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);
! 214: [(_8)/(x),[_8]]
! 215:
! 216: [350] D = diff_op_appell4(0,0,3,0,[x,y])$
! 217: [351] diff_rat_solve(D,x^2,2,[x,y]);
! 218: [(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]]
1.1 takayama 219: @end example
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