Annotation of OpenXM/src/asir-contrib/packages/doc/Diff.texi, Revision 1.6
1.6 ! okutani 1: @c $OpenXM: OpenXM/src/asir-contrib/packages/doc/Diff.texi,v 1.5 2000/01/31 11:01:00 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
1.6 ! okutani 7: $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.
! 8: $B%j%9%H$K$h$kI=8=$O<!$N$h$&$K$J$j$^$9(B.
1.1 takayama 9: $$ [ [f_{\alpha},[\alpha_{1},\ldots,\alpha_{n}]],\ldots ] $$
10: $B$3$l$O(B
11: $$ \sum_{\alpha}f_{\alpha}\partial^{\alpha} $$
12: $B$H$$$&0UL#$G$9(B. $B@~7?JPHyJ,J}Dx<07O(B
13: $$ (\sum_{\alpha^{(i)}}f_{\alpha^{(i)}}\partial^{\alpha^{(i)}})\bullet u = 0 \quad (i = 1,\ldots,s) $$
14: $B$J$I$N$h$&$KJ#?t$NHyJ,:nMQAG$rI=8=$9$k$H$-$OHyJ,:nMQAG$N%j%9%H$r;H$$$^$9(B.
15: $$ [ [ [f_{\alpha^{(1)}},[\alpha_{1}^{(1)},\ldots,\alpha_{n}^{(1)}]],\ldots ],\ldots,[ [f_{\alpha^{(s)}},[\alpha_{1}^{(s)},\ldots,\alpha_{n}^{(s)}]],\ldots ] ] $$
1.6 ! okutani 16: $BNc$($PHyJ,:nMQAG(B$x \partial_{x} + y \partial_{y} + 1$$B$N>l9g$O(B
1.1 takayama 17: $$ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ] $$
1.6 ! okutani 18: $B$H$J$j$^$9(B. $B$^$?HyJ,:nMQAG$N%j%9%H$G(B$x \partial_{x} + y \partial_{y} + 1, {\partial_{x}}^{2} + {\partial_{y}}^{2}$$B$rI=$9$H(B
1.1 takayama 19: $$ [ [ [x,[1,0]],[y,[0,1]],[1,[0,0]] ],[ [1,[2,0]],[1,[0,2]] ] ] $$
1.6 ! okutani 20: $B$H$J$j$^$9(B. $B$^$?$3$NI=8=K!$r;H$&$H$-$OJQ?t%j%9%H$r>o$K0U<1$7$F$$$kI,MW$,$"$j$^$9(B.
! 21: $B<!$KB?9`<0$K$h$kI=8=$K$D$$$F=R$Y$^$9(B. $BJQ?t(B$x$$B$KBP$9$kHyJ,$O(B$dx$$B$GI=8=$5$l$^$9(B.
! 22: $BNc$($P(B$x \partial_{x} + y \partial_{y} + 1$$B$K$D$$$F$O(B
! 23: $$ x*dx+y*dy+1 $$
! 24: $B$HI=8=$5$l$^$9(B.
1.1 takayama 25: @end tex
26: @menu
27: @c * diff_op_hg1::
28: @c * diff_op_appell1::
29: @c * diff_op_appell2::
30: @c * diff_op_appell3::
31: * diff_op_appell4::
32: @c * diff_op_selberg2::
33: @c * diff_op_gkz::
1.6 ! okutani 34: * diff_op_tosm1::
! 35: * diff_op_toasir::
! 36: * diff_op_fromasir::
1.1 takayama 37: * diff_act::
38: @c * diff_act_hg1::
39: @c * diff_act_appell1::
40: @c * diff_act_appell2::
41: @c * diff_act_appell3::
42: * diff_act_appell4::
43: @c * diff_act_selberg2::
44: @c * diff_act_gkz::
45: * diff_poly_solve::
1.5 okutani 46: * diff_poly_solve_hg1::
1.1 takayama 47: @c * diff_poly_solve_appell1::
48: @c * diff_poly_solve_appell2::
49: @c * diff_poly_solve_appell3::
1.2 okutani 50: * diff_poly_solve_appell4::
1.1 takayama 51: @c * diff_poly_solve_selberg2::
52: @c * diff_poly_solve_gkz::
1.3 okutani 53: * diff_rat_solve::
1.2 okutani 54: @c * diff_pseries_appell4::
1.1 takayama 55: @end menu
56:
57: @node diff_op_appell4,,, Differential equations
58: @subsection @code{diff_op_appell4}
59: @findex diff_op_appell4
60: @table @t
61: @item diff_op_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
62: :: appell $B$N(B F_4 $B$rNm2=$9$kHyJ,:nMQAG$r@8@.$7$^$9(B.
63: @end table
64: @table @var
65: @item return
66: $B%j%9%H(B
67: @item a, b, c1, c2
68: $BM-M}<0(B
69: @item V
70: $B%j%9%H(B
71: @end table
72: @itemize @bullet
73: @item @code{diff_op_appell4}$B$NNc(B.
74: @end itemize
75: @example
76: [298] diff_op_appell4(a,b,c1,c2,[x,y]);
77: [ [ [-x^2+x,[2,0]], [-2*y*x,[1,1]], [-y^2,[0,2]],
78: [(-a-b-1)*x+c1,[1,0]], [(-a-b-1)*y,[0,1]], [-b*a,[0,0]] ],
79: [ [-y^2+y,[0,2]], [-2*y*x,[1,1]], [-x^2,[2,0]],
80: [(-a-b-1)*y+c2,[0,1]], [(-a-b-1)*x,[1,0]], [-b*a,[0,0]] ] ]
81: @end example
82:
1.6 ! okutani 83: @node diff_op_tosm1,,, Differential equations
! 84: @subsection @code{diff_op_tosm1}
! 85: @findex diff_op_tosm1
! 86: @table @t
! 87: @item diff_op_tosm1(@var{LL},@var{V})
! 88: :: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$r(B sm1 $B7A<0$KJQ49$7$^$9(B.
! 89: @end table
! 90: @table @var
! 91: @item return
! 92: $B%j%9%H(B
! 93: @item LL
! 94: $B%j%9%H(B
! 95: @item V
! 96: $B%j%9%H(B
! 97: @end table
! 98: @itemize @bullet
! 99: @item $BHyJ,:nMQAG$N78?t$O@0?tB?9`<0$KJQ49$5$l$^$9(B.
! 100: @item @code{diff_op_tosm1}$B$NNc(B
! 101: @end itemize
! 102: @example
! 103: [299] diff_op_tosm1([[[x,[2,0]],[-1,[0,0]]],
! 104: [[y,[0,2]],[-1,[0,0]]]],[x,y]);
! 105: [ + ( + (1) x) dx^2 + ( + (-1)), + ( + (1) y) dy^2 + ( + (-1))]
! 106:
! 107: [300] diff_op_tosm1([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
! 108: [[1,[2,0]],[1,[0,2]]]],[x,y]);
! 109: [ + ( + (1) x) dx + ( + (1) y) dy + ( + (1)), + ( + (1)) dx^2 + ( + (1)) dy^2]
! 110:
! 111: [301] diff_op_tosm1([[[1/2,[1,0]],[1,[0,0]]],
! 112: [[1/3,[0,1]],[1/4,[0,0]]]],[x,y]);
! 113: [ + ( + (6)) dx + ( + (12)), + ( + (4)) dy + ( + (3))]
! 114:
! 115: [302] diff_op_tosm1([[[1/2*x,[1,0]],[1,[0,0]]],
! 116: [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
! 117: [ + ( + (6) x) dx + ( + (12)), + ( + (4) y) dy + ( + (3))]
! 118: @end example
! 119:
! 120: @node diff_op_toasir,,, Differential equations
! 121: @subsection @code{diff_op_toasir}
! 122: @findex diff_op_toasir
! 123: @table @t
! 124: @item diff_op_toasir(@var{LL},@var{V})
! 125: :: $B%j%9%H7A<0$NHyJ,:nMQAG%j%9%H(B @var{LL} $B$r(B @code{asir} $B$NB?9`<0$KJQ49$7$^$9(B.
! 126: @end table
! 127: @table @var
! 128: @item return
! 129: $B%j%9%H(B
! 130: @item LL
! 131: $B%j%9%H(B
! 132: @item V
! 133: $B%j%9%H(B
! 134: @end table
! 135: @itemize @bullet
! 136: @item @code{diff_op_toasir}$B$NNc(B
! 137: @end itemize
! 138: @example
! 139: [303] diff_op_toasir([[[1/2*x,[1,0]],[1,[0,0]]],
! 140: [[1/3*y,[0,1]],[1/4,[0,0]]]],[x,y]);
! 141: [1/2*x*dx+1,1/3*y*dy+1/4]
! 142:
! 143: [304] diff_op_toasir([[[x,[1,0]],[y,[0,1]],[1,[0,0]]],
! 144: [[1,[2,0]],[1,[0,2]]]],[x,y]);
! 145: [x*dx+y*dy+1,dx^2+dy^2]
! 146: @end example
! 147:
! 148: @node diff_op_fromasir,,, Differential equations
! 149: @subsection @code{diff_op_fromasir}
! 150: @findex diff_op_fromasir
! 151: @table @t
! 152: @item diff_op_fromasir(@var{D_list},@var{V})
! 153: :: @code{asir} $B$NB?9`<0$+$i%j%9%H7A<0$NHyJ,:nMQAG%j%9%H$KJQ49$7$^$9(B.
! 154: @end table
! 155: @table @var
! 156: @item return
! 157: $B%j%9%H(B
! 158: @item D_list
! 159: $B%j%9%H(B
! 160: @item V
! 161: $B%j%9%H(B
! 162: @end table
! 163: @itemize @bullet
! 164: @item @code{diff_op_fromasir}$B$NNc(B
! 165: @end itemize
! 166: @example
! 167: [305] diff_op_fromasir([1/2*x*dx+1,1/3*y*dy+1/4],[x,y]);
! 168: [[[1/2*x,[1,0]],[1,[0,0]]],[[1/3*y,[0,1]],[1/4,[0,0]]]]
! 169:
! 170: [306] diff_op_fromasir([x*dx+y*dy+1,dx^2+dy^2],[x,y]);
! 171: [[[x,[1,0]],[y,[0,1]],[1,[0,0]]],[[1,[2,0]],[1,[0,2]]]]
! 172: @end example
! 173:
1.1 takayama 174: @node diff_act,,, Differential equations
175: @subsection @code{diff_act}
176: @findex diff_act
177: @table @t
178: @item diff_act(@var{L},@var{F},@var{V})
179: :: $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.
180: @end table
181: @table @var
182: @item return
183: $BM-M}<0(B
184: @item L
1.6 ! okutani 185: $B%j%9%H(B or $BB?9`<0(B
1.1 takayama 186: @item F
187: $BM-M}<0(B
188: @item V
189: $B%j%9%H(B
190: @end table
191: @itemize @bullet
192: @item @code{diff_act}$B$NNc(B
193: @end itemize
194: @example
195: [302] diff_act([[1,[2]]],x^3+x^2+x+1,[x]);
196: 6*x+2
197:
198: [303] diff_act([[1,[1,0]],[1,[0,1]]],x^2+y^2,[x,y]);
199: 2*x+2*y
1.6 ! okutani 200:
! 201: [349] diff_act(x*dx+y*dy, x^2+x*y+y^2, [x,y]);
! 202: 2*x^2+2*y*x+2*y^2
1.1 takayama 203: @end example
204:
205: @node diff_act_appell4,,, Differential equations
206: @subsection @code{diff_act_appell4}
207: @findex diff_act_appell4
208: @table @t
209: @item diff_act_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{F},@var{V})
210: :: $BHyJ,:nMQAG(B @code{diff_op_appell4} $B$rM-M}<0(B @var{F} $B$K:nMQ$5$;$k(B.
211: @end table
212: @table @var
213: @item return
214: $B%j%9%H(B
215: @item a, b, c1, c2
216: $BM-M}<0(B
217: @item F
218: $BM-M}<0(B
219: @item V
220: $B%j%9%H(B
221: @end table
222: @itemize @bullet
223: @item @code{diff_act_appell4}$B$NNc(B
224: @end itemize
225: @example
226: [303] diff_act_appell4(1,0,1,1,x^2+y^2,[x,y]);
227: [-6*x^2+4*x-6*y^2,-6*x^2-6*y^2+4*y]
228:
229: [304] diff_act_appell4(0,0,1,1,x^2+y^2,[x,y]);
230: [-4*x^2+4*x-4*y^2,-4*x^2-4*y^2+4*y]
231:
232: [305] diff_act_appell4(-2,-2,-1,-1,x^2+y^2,[x,y]);
233: [0,0]
234: @end example
235:
236: @node diff_poly_solve,,, Differential equations
237: @subsection @code{diff_poly_solve}
238: @findex diff_poly_solve
239: @table @t
240: @item diff_poly_solve(@var{LL},@var{N},@var{V})
241: :: $BM?$($i$l$?@~7?HyJ,J}Dx<07O$N(B @var{N} $B<!0J2<$NB?9`<02r$r5a$a$k(B.
242: @end table
243: @table @var
244: @item return
245: $B%j%9%H(B
246: @item LL
247: $B%j%9%H(B
248: @item N
249: $B@0?t(B
250: @item V
251: $B%j%9%H(B
252: @end table
253: @itemize @bullet
254: @item @code{diff_poly_solve}$B$NNc(B.
255: @end itemize
256: @example
257: [297] diff_poly_solve([[[x,[1,0]],[-1,[0,0]]],[[y,[0,1]],[-1,[0,0]]]],5,[x,y]);
258: [_4*y*x,[_4]]
259:
260: [298] diff_poly_solve([[[x,[1,0]],[-2,[0,0]]],[[y,[0,1]],[-2,[0,0]]]],5,[x,y]);
261: [_33*y^2*x^2,[_33]]
1.6 ! okutani 262:
! 263: [356] diff_poly_solve([x*dx+y*dy-3,dx+dy],4,[x,y]);
! 264: [-_126*x^3+3*_126*y*x^2-3*_126*y^2*x+_126*y^3,[_126]]
1.2 okutani 265: @end example
266:
1.5 okutani 267: @node diff_poly_solve_hg1,,, Differential equations
268: @subsection @code{diff_poly_solve_hg1}
269: @findex diff_poly_solve_hg1
270: @table @t
271: @item diff_poly_solve_hg1(@var{a},@var{b},@var{c},@var{V})
272: :: $B%,%&%9$ND64v2?HyJ,J}Dx<0$NB?9`<02r$r5a$a$k(B.
273: @end table
274: @table @var
275: @item return
276: $B%j%9%H(B
277: @item a, b, c
278: $BM-M}<0(B
279: @item V
280: $B%j%9%H(B
281: @end table
282: @itemize @bullet
283: @item @code{diff_poly_solve_hg1}$B$NNc(B.
284: @end itemize
285: @example
286: [334] diff_poly_solve_hg1(-3,-6,-5,[x]);
287: [_1*x^6-2*_0*x^3+9/2*_0*x^2-18/5*_0*x+_0,[_0,_1]]
288:
289: [335] diff_poly_solve_hg1(-3,-6,-7,[x]);
290: [-4/7*_2*x^3+15/7*_2*x^2-18/7*_2*x+_2,[_2]]
291: @end example
292:
1.2 okutani 293: @node diff_poly_solve_appell4,,, Differential equations
294: @subsection @code{diff_poly_solve_appell4}
295: @findex diff_poly_solve_appell4
296: @table @t
1.5 okutani 297: @item diff_poly_solve_appell4(@var{a},@var{b},@var{c1},@var{c2},@var{V})
298: :: F_4$B$,$_$?$9@~7?HyJ,J}Dx<07O$NB?9`<02r$r5a$a$k(B.
1.2 okutani 299: @end table
300: @table @var
301: @item return
302: $B%j%9%H(B
303: @item a, b, c1, c2
304: $BM-M}<0(B
305: @item V
306: $B%j%9%H(B
307: @end table
308: @itemize @bullet
309: @item @code{diff_poly_solve_appell4}$B$NNc(B.
310: @end itemize
311: @example
1.5 okutani 312: [299] diff_poly_solve_appell4(-3,1,-1,-1,[x,y]);
1.2 okutani 313: [-_26*x^3+(3*_26*y+_26)*x^2+3*_24*y^2*x-_24*y^3+_24*y^2,[_24,_26]]
314:
1.5 okutani 315: [300] diff_poly_solve_appell4(-3,1,1,-1,[x,y]);
1.2 okutani 316: [-3*_45*y^2*x-_45*y^3+_45*y^2,[_45]]
1.3 okutani 317: @end example
318:
319: @node diff_rat_solve,,, Differential equations
320: @subsection @code{diff_rat_solve}
321: @findex diff_rat_solve
322: @table @t
323: @item diff_rat_solve(@var{LL},@var{Dn},@var{N},@var{V})
324: :: $BM?$($i$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.
325: @end table
326: @table @var
327: @item return
328: $B%j%9%H(B
329: @item LL
330: $B%j%9%H(B
331: @item Dn
332: $BM-M}<0(B
333: @item N
334: $B@0?t(B
335: @item V
336: $B%j%9%H(B
337: @end table
338: @itemize @bullet
339: @item @code{diff_rat_solve}$B$NNc(B.
340: @end itemize
341: @example
1.4 okutani 342: [333] diff_rat_solve([[[x,[1]],[1,[0]]]],x,1,[x]);
343: [(_8)/(x),[_8]]
1.6 ! okutani 344:
! 345: [361] diff_rat_solve([x*(1-x)*dx^2+(1-3*x)*dx-1],1-x,2,[x]);
! 346: [(_180)/(-x+1),[_180]]
1.4 okutani 347:
348: [350] D = diff_op_appell4(0,0,3,0,[x,y])$
349: [351] diff_rat_solve(D,x^2,2,[x,y]);
350: [(_118*x^2-_114*y*x+1/2*_114*y^2+_114*y)/(x^2),[_114,_118]]
1.1 takayama 351: @end example
FreeBSD-CVSweb <freebsd-cvsweb@FreeBSD.org>
![[BACK]](/icons/cvsweb/back.gif){kind=icon}
Return to Diff.texi CVS log ![[TXT]](/icons/cvsweb/text.gif){kind=icon}
![[DIR]](/icons/cvsweb/dir.gif){kind=icon}
Up to [local] / OpenXM / src / asir-contrib / packages / doc