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%!PS-Adobe-2.0 %%Creator: dvips 5.511 Copyright 1986, 1993 Radical Eye Software %%Title: /export/salvy/Tex/Sem/Sem98-99/All.dvi %%CreationDate: Mon Feb 7 16:44:24 2000 %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: /usr/local/lib/tex/bin/dvips -pp 43-48 -o weil.ps /export/salvy/Tex/Sem/Sem98-99/All.dvi %DVIPSSource: TeX output 1999.12.15:1241 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N} B /TR{translate}N /isls false N /vsize 11 72 mul N /@rigin{isls{[0 -1 1 0 0 0] concat}if 72 Resolution div 72 VResolution div neg scale isls{Resolution hsize -72 div mul 0 TR}if Resolution VResolution vsize -72 div 1 add mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn 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Fi(y)279 1795 y Ff(0)277 1824 y Fh(i)291 1811 y Fi(=y)336 1818 y Fh(i)370 1811 y Fr(is)20 b(the)g(logarithmic)h(deriv)m(ativ)o(e)f(of)g(the)f(pro)q(duct)1275 1777 y Fg(Q)1317 1825 y Fh(i)1339 1811 y Fi(y)1361 1818 y Fh(i)1376 1811 y Fr(.)33 b(A)20 b(necessary)f(and)h(su\016cien)o(t)0 1865 y(condition)g(for)d(the)h(existence)i(of)e(a)g(p)q(olynomial)i Fi(P)k Fr(of)18 b(degree)h Fi(m)f Fr(ab)q(o)o(v)o(e,)g(whic)o(h)h(describ)q (es)h(the)e(liouvilli)q(an)0 1919 y(solutions)h(of)f Fi(L)h Fr(is)f(that)g(there)h(exists)g(a)f(p)q(olynomial)i(of)e(degree)h Fi(m)f Fr(in)h(solutions)g(of)f Fi(L)h Fr(whose)f(logarithmic)0 1973 y(deriv)m(ativ)o(e)h(is)e(rational,)h(and)f(whic)o(h)h(is)g(the)g(pro)q (duct)f(of)g(linear)h(factors.)25 b(More)17 b(sp)q(eci\014cally)l(,)k(for)16 b(a)h(solution)0 2027 y(basis)f(\()p Fi(z)151 2034 y Fn(1)170 2027 y Fi(;)8 b(:)g(:)g(:)d(;)j(z)293 2034 y Fh(m)326 2027 y Fr(\))14 b(of)h Fi(L)g Fr(the)h(pro)q(duct)705 1993 y Fg(Q)748 2041 y Fh(i)770 2027 y Fi(y)792 2034 y Fh(i)821 2027 y Fr(is)g(searc)o(hed)f (for)g(under)h(the)f(form)1435 1993 y Fg(Q)1478 2041 y Fh(i)1492 2027 y Fr(\()p Fi(c)1530 2034 y Fh(i;)p Fn(1)1571 2027 y Fi(z)1592 2034 y Fn(1)1621 2027 y Fr(+)c Fe(\001)d(\001)g(\001)g Fr(+)j Fi(c)1796 2034 y Fh(i;m)1850 2027 y Fi(z)1871 2034 y Fh(m)1905 2027 y Fr(\).)50 2081 y(The)k(searc)o(h)g(for)g(liouvilli)q(an)j(solutions)e (therefore)f(reduces)h(to)f(the)g(searc)o(h)g(for)g(exp)q(onen)o(tial)h (solutions.)21 b(T)l(o)0 2135 y(this)15 b(end,)f(the)h(presen)o(t)f(w)o(ork)f (allo)o(ws)i(to)e(a)o(v)o(oid)h(computing)h(the)f(equation)h(for)f(the)g (symmetric)h(p)q(o)o(w)o(er,)e(whic)o(h)0 2189 y(is)j(to)q(o)e(large,)h(but)g (prefers)h(a)f(more)f(compact)h(represen)o(tation.)434 2306 y(4.)26 b Fq(F)l(actorization)19 b(and)f(Alternate)g(P)o(o)o(w)o(er)d (Solutions)50 2387 y Fr(As)f(another)g(t)o(ypical)h(example,)g(let)g(us)g (consider)g(the)g(searc)o(h)f(for)g(a)g(righ)o(t-hand)g(factor)g Fi(H)i Fr(=)d Fi(@)1721 2371 y Fn(2)1749 2387 y Fe(\000)c Fi(b)1813 2394 y Fn(1)1832 2387 y Fi(@)i Fe(\000)e Fi(b)1931 2394 y Fn(0)0 2441 y Fr(of)14 b(order)f(2)h(of)f(the)h(op)q(erator)g Fi(L)e Fr(=)h Fi(@)633 2425 y Fn(4)660 2441 y Fe(\000)8 b Fi(a)727 2448 y Fn(2)747 2441 y Fi(@)774 2425 y Fn(2)801 2441 y Fe(\000)g Fi(a)868 2448 y Fn(1)888 2441 y Fi(@)i Fe(\000)e Fi(a)989 2448 y Fn(0)1022 2441 y Fr(of)14 b(order)g(4.)19 b(F)l(or)13 b(an)o(y)h(solution)h (basis)f(\()p Fi(y)1732 2448 y Fn(1)1752 2441 y Fi(;)8 b(y)1795 2448 y Fn(2)1814 2441 y Fr(\))13 b(of)h Fi(H)t Fr(,)0 2495 y(the)h(op)q(erator)g Fi(H)j Fr(is)e(giv)o(en)g(b)o(y)f(the)g(determinan)o (tal)h(represen)o(tation)95 2629 y Fi(H)t Fr(\()p Fi(y)r Fr(\))11 b(=)256 2563 y Fg(\014)256 2590 y(\014)256 2618 y(\014)256 2645 y(\014)292 2601 y Fi(y)314 2608 y Fn(1)376 2601 y Fi(y)398 2608 y Fn(2)292 2655 y Fi(y)316 2639 y Ff(0)314 2667 y Fn(1)376 2655 y Fi(y)400 2639 y Ff(0)398 2667 y Fn(2)439 2563 y Fg(\014)439 2590 y(\014)439 2618 y(\014)439 2645 y(\014)454 2576 y Ff(\000)p Fn(1)509 2536 y Fg(\014)509 2563 y(\014)509 2590 y(\014)509 2618 y(\014)509 2645 y(\014)509 2672 y(\014)555 2574 y Fi(y)56 b(y)655 2581 y Fn(1)719 2574 y Fi(y)741 2581 y Fn(2)549 2628 y Fi(y)573 2612 y Ff(0)633 2628 y Fi(y)657 2612 y Ff(0)655 2640 y Fn(1)719 2628 y Fi(y)743 2612 y Ff(0)741 2640 y Fn(2)544 2682 y Fi(y)568 2666 y Ff(00)631 2682 y Fi(y)655 2666 y Ff(00)653 2694 y Fn(1)718 2682 y Fi(y)742 2666 y Ff(0)o(0)740 2694 y Fn(2)784 2536 y Fg(\014)784 2563 y(\014)784 2590 y(\014)784 2618 y(\014)784 2645 y(\014)784 2672 y(\014)811 2629 y Fr(=)13 b Fi(y)883 2610 y Ff(00)915 2629 y Fe(\000)965 2598 y Fi(!)993 2605 y Fn(0)p Fh(;)p Fn(2)p 965 2618 76 2 v 965 2660 a Fi(!)993 2667 y Fn(0)p Fh(;)p Fn(1)1045 2629 y Fi(y)1069 2610 y Ff(0)1091 2629 y Fr(+)1142 2598 y Fi(!)1170 2605 y Fn(1)p Fh(;)p Fn(2)p 1142 2618 V 1142 2660 a Fi(!)1170 2667 y Fn(0)p Fh(;)p Fn(1)1222 2629 y Fi(y)47 b Fr(where)f Fi(!)1481 2636 y Fh(i;j)1534 2629 y Fr(=)1582 2549 y Fg(\014)1582 2577 y(\014)1582 2604 y(\014)1582 2631 y(\014)1582 2658 y(\014)1620 2601 y Fi(y)1644 2577 y Fn(\()p Fh(i)p Fn(\))1642 2614 y(1)1731 2601 y Fi(y)1755 2577 y Fn(\()p Fh(i)p Fn(\))1753 2614 y(2)1618 2666 y Fi(y)1642 2642 y Fn(\()p Fh(j)r Fn(\))1640 2679 y(1)1729 2666 y Fi(y)1753 2642 y Fn(\()p Fh(j)r Fn(\))1751 2679 y(2)1820 2549 y Fg(\014)1820 2577 y(\014)1820 2604 y(\014)1820 2631 y(\014)1820 2658 y(\014)1842 2629 y Fi(:)p eop %%Page: 45 3 45 2 bop 1915 22 a Fn(45)0 120 y Fr(T)l(o)16 b(obtain)h(a)g(factor)f(of)g (order)g(2,)h(w)o(e)f(no)o(w)g(searc)o(h)h(for)f(an)h(exp)q(onen)o(tial)h (solution)f(and)g(sho)o(w)f(that)g(it)h(can)g(b)q(e)0 174 y(in)o(terpreted)f (as)f(a)g(determinan)o(t)g Fi(!)612 181 y Fn(0)p Fh(;)p Fn(1)659 174 y Fr(.)20 b(Let)c Fi(A)f Fr(b)q(e)h(the)f(companion)h(matrix)151 334 y Fi(A)c Fr(=)245 215 y Fg(0)245 295 y(B)245 323 y(B)245 352 y(@)316 252 y Fr(0)63 b(1)f(0)52 b(0)316 306 y(0)63 b(0)f(1)52 b(0)316 360 y(0)63 b(0)f(0)52 b(1)306 414 y Fi(a)330 421 y Fn(0)391 414 y Fi(a)415 421 y Fn(1)476 414 y Fi(a)500 421 y Fn(2)562 414 y Fr(0)605 215 y Fg(1)605 295 y(C)605 323 y(C)605 352 y(A)652 334 y Fi(;)99 b Fr(and)15 b(let)91 b Fi(Y)23 b Fr(=)1090 215 y Fg(0)1090 295 y(B)1090 323 y(B)1090 352 y(@)1166 252 y Fi(y)1161 306 y(y)1185 290 y Ff(0)1156 360 y Fi(y)1180 344 y Ff(0)o(0)1151 414 y Fi(y)1175 398 y Ff(0)o(00)1226 215 y Fg(1)1226 295 y(C)1226 323 y(C)1226 352 y(A)1274 334 y Fi(;)98 b Fr(so)15 b(that)90 b Fi(Y)1651 315 y Ff(0)1676 334 y Fr(=)13 b Fi(AY)s(:)0 497 y Fr(Let)f(us)h(in)o(tro)q(duce)g(the)f(v)o(ector)f Fi(Z)16 b Fr(=)d(\()p Fi(!)682 504 y Fn(0)p Fh(;)p Fn(1)729 497 y Fi(;)8 b(!)778 504 y Fn(0)p Fh(;)p Fn(2)825 497 y Fi(;)g(!)874 504 y Fn(0)p Fh(;)p Fn(3)921 497 y Fi(;)g(!)970 504 y Fn(1)p Fh(;)p Fn(2)1016 497 y Fi(;)g(!)1065 504 y Fn(1)p Fh(;)p Fn(3)1112 497 y Fi(;)g(!)1161 504 y Fn(2)p Fh(;)p Fn(3)1208 497 y Fr(\))1226 480 y Fh(T)1253 497 y Fr(.)19 b(In)12 b(view)h(of)f(their)g(de\014nition,)i (the)f Fi(!)1910 504 y Fh(i;j)0 551 y Fr(satisfy)g(di\013eren)o(tial)i (relations)f(lik)o(e)h Fi(!)661 534 y Ff(0)659 563 y Fn(0)p Fh(;)p Fn(1)719 551 y Fr(=)e Fi(!)795 558 y Fn(0)p Fh(;)p Fn(2)842 551 y Fr(,)27 b Fi(!)912 534 y Ff(0)910 563 y Fn(0)p Fh(;)p Fn(3)970 551 y Fr(=)13 b Fi(!)1046 558 y Fn(1)p Fh(;)p Fn(3)1101 551 y Fr(+)7 b Fi(a)1167 558 y Fn(0)1186 551 y Fi(!)1214 558 y Fn(0)p Fh(;)p Fn(0)1269 551 y Fr(+)g Fi(a)1335 558 y Fn(1)1355 551 y Fi(!)1383 558 y Fn(0)p Fh(;)p Fn(1)1437 551 y Fr(+)g Fi(a)1503 558 y Fn(2)1523 551 y Fi(!)1551 558 y Fn(0)p Fh(;)p Fn(2)1598 551 y Fr(,)14 b(and)f(so)g(on.)20 b(F)l(rom)0 605 y(them,)15 b(w)o(e)g(\014nd)h(a)e(matrix)270 815 y(\003)302 822 y Fn(2)321 815 y Fr(\()p Fi(A)p Fr(\))e(=)451 642 y Fg(0)451 722 y(B)451 749 y(B)451 776 y(B)451 803 y(B)451 831 y(B)451 858 y(B)451 887 y(@)540 679 y Fr(0)98 b(1)80 b(0)f(0)70 b(0)41 b(0)540 733 y(0)98 b(0)80 b(1)f(1)70 b(0)41 b(0)529 787 y Fi(a)553 794 y Fn(1)650 787 y Fi(a)674 794 y Fn(2)764 787 y Fr(0)79 b(0)70 b(1)41 b(0)540 841 y(0)98 b(0)80 b(0)f(0)70 b(1)41 b(0)512 895 y Fe(\000)p Fi(a)571 902 y Fn(0)661 895 y Fr(0)69 b Fi(a)777 902 y Fn(2)866 895 y Fr(0)h(0)41 b(1)540 949 y(0)69 b Fe(\000)p Fi(a)691 956 y Fn(0)764 949 y Fr(0)51 b Fe(\000)p Fi(a)897 956 y Fn(1)959 949 y Fr(0)41 b(0)1067 642 y Fg(1)1067 722 y(C)1067 749 y(C)1067 776 y(C)1067 803 y(C)1067 831 y(C)1067 858 y(C)1067 887 y(A)1205 815 y Fr(suc)o(h)15 b(that)g Fi(Z)1440 796 y Ff(0)1464 815 y Fr(=)e(\003)1544 822 y Fn(2)1564 815 y Fr(\()p Fi(A)p Fr(\))p Fi(Z)s Fr(.)0 1027 y(Again,)f(w)o(e)g(then)g(only)g(lo)q(ok)g(for)f (exp)q(onen)o(tial)i(solutions)f Fi(Z)j Fr(of)c(the)h(matrix)f(\003)1334 1034 y Fn(2)1354 1027 y Fr(\()p Fi(A)p Fr(\),)g(whic)o(h)i(is)f(easy)f(to)g (construct)0 1081 y(and)22 b(con)o(tains)h(more)e(information)i(than)f(the)g (usual)h(single)g(auxiliary)h(equation)e(used)h(for)f(factorization.)0 1135 y(Finally)l(,)17 b(one)e(has)g(to)g(c)o(hec)o(k)g(that)g(the)g(solution) h Fi(Z)j Fr(is)d(a)f(determinan)o(t.)20 b(F)l(or)15 b(this,)g(a)g(necessary)h (and)f(su\016cien)o(t)0 1189 y(condition)h(is)g(the)f(Pl)q(\177)-24 b(uc)o(k)o(er)16 b(relation,)g(whic)o(h)g(here)f(simply)i(reduces)f(to)e Fi(!)1286 1196 y Fn(0)p Fh(;)p Fn(1)1334 1189 y Fi(!)1362 1196 y Fn(2)p Fh(;)p Fn(3)1419 1189 y Fe(\000)d Fi(!)1493 1196 y Fn(0)p Fh(;)p Fn(2)1540 1189 y Fi(!)1568 1196 y Fn(1)p Fh(;)p Fn(3)1626 1189 y Fr(+)f Fi(!)1699 1196 y Fn(0)p Fh(;)p Fn(3)1747 1189 y Fi(!)1775 1196 y Fn(1)p Fh(;)p Fn(2)1835 1189 y Fr(=)j(0.)50 1243 y(T)l(o)h(rephrase)h(the)f(metho)q(d)h(in)g(a)f(more)g(formal)h(w)o(a)o (y)l(,)e(in)o(tro)q(duce)i Fi(V)10 b Fr(,)k(the)h(solution)g(space)g(of)f Fi(L)p Fr(.)19 b(The)c(searc)o(h)0 1297 y(for)h Fi(Z)21 b Fr(is)c(indeed)i(a) e(searc)o(h)g(for)f(ob)s(jects)g(in)i(the)f(2-exterior)g(p)q(o)o(w)o(er)g (\003)1234 1304 y Fn(2)1253 1297 y Fr(\()p Fi(V)9 b Fr(\),)17 b(i.e.,)g(the)h(v)o(ector)e(space)h(of)g(linear)0 1351 y(com)o(bination)k(of) f(formal)f(2-exterior)i(pro)q(ducts)f Fi(v)15 b Fe(^)f Fi(w)q Fr(,)41 b(\()p Fi(v)r(;)8 b(w)q Fr(\))19 b Fe(2)i Fi(V)1264 1334 y Fn(2)1284 1351 y Fr(,)g(whic)o(h)g(satisfy)f(the)g(rule)h Fi(w)14 b Fe(^)g Fi(v)23 b Fr(=)0 1405 y Fe(\000)p Fi(v)10 b Fe(^)f Fi(w)q Fr(.)19 b(Pure)c(exterior)f(pro)q(duct)g Fi(u)9 b Fe(^)f Fi(v)16 b Fr(are)e(in)o(terpreted)h(as)f(determinan)o(ts.)20 b(The)14 b(searc)o(h)g(for)g Fi(Z)j Fr(is)e(therefore)0 1459 y(equiv)m(alen)o(t)f(to)d(the)h(searc)o(h)g(for)f(a)h(pure)g(exterior)g(pro)q (duct)g Fi(!)1036 1466 y Fn(0)p Fh(;)p Fn(1)1096 1459 y Fe(2)h Fr(\003)1171 1442 y Fn(2)1190 1459 y Fr(\()p Fi(V)d Fr(\))h(suc)o(h)i(that)e (the)h(1-dimensional)i(v)o(ector)0 1513 y(space)h Fi(C)s(!)185 1520 y Fn(0)p Fh(;)p Fn(1)248 1513 y Fr(is)h(stable)f(under)h(the)f(action)h (of)e(the)i(di\013eren)o(tial)g(Galois)g(group)e(of)h Fi(L)p Fr(.)50 1567 y(Here)e(the)h(searc)o(h)f(for)g(a)g(second-order)h(righ)o (t-hand)g(factor)e(of)h(a)g(fourth-order)g(equation)h(has)f(b)q(een)i (reduced)0 1621 y(to)g(solving)h(a)e(\\simpler")i(system)f(of)g(six)h (\014rst-order)e(equations.)308 1720 y(5.)26 b Fq(Mo)q(dule)18 b(and)g(Dual)h(Mo)q(dule)f(Asso)q(ciated)g(with)g(an)g(Op)q(erator)50 1801 y Fr(As)c(an)f(imp)q(ortan)o(t)h(to)q(ol)f(for)h(the)f(study)h(of)g(a)f (linear)i(di\013eren)o(tial)g(op)q(erator)e Fi(L)p Fr(,)h(one)g(classically)i (asso)q(ciates)d(a)0 1855 y(canonical)h(mo)q(dule)f(in)g(the)g(follo)o(wing)g (w)o(a)o(y)l(.)18 b(F)l(or)11 b Fi(L)h Fr(in)i(the)e(algebra)g Fi(k)q Fr([)p Fi(@)s Fr(])f(of)h(linear)i(di\013eren)o(tial)f(op)q(erators)f (with)0 1909 y(co)q(e\016cien)o(ts)18 b(in)f(a)g(\014eld)h Fi(k)q Fr(,)e(one)h(considers)h(the)f(quotien)o(t)g Fi(M)j Fr(=)15 b Fi(k)q Fr([)p Fi(@)s Fr(])p Fi(=k)q Fr([)p Fi(@)s Fr(])p Fi(L)g Fr(of)h Fi(k)q Fr([)p Fi(@)s Fr(])g(b)o(y)g(its)h(left)g(ideal) h Fi(k)q Fr([)p Fi(@)s Fr(])p Fi(L)p Fr(.)0 1963 y(The)h(left)g(mo)q(dule)g Fi(M)24 b Fr(can)18 b(b)q(e)i(view)o(ed)f(as)f(the)h(mo)q(dule)g Fi(k)q Fr([)p Fi(@)s Fr(])p Fi(y)h Fr(generated)e(b)o(y)h(a)f(generic)h (solution)g Fi(y)i Fr(of)d(the)0 2017 y(op)q(erator)e Fi(L)p Fr(.)26 b(Linear)18 b(constructs)f(on)g(and)g(b)q(et)o(w)o(een)h(solution)g (spaces)f(of)g(op)q(erators,)f(lik)o(e)i(\(direct)g(or)f(usual\))0 2071 y(sums,)f(\(symmetric)g(or)g(exterior)g(or)f(usual)i(comm)o(utativ)o (e\))e(pro)q(ducts,)i(\(inde\014nite\))g(in)o(tegration,)f(and)h(so)e(on,)0 2125 y(corresp)q(ond)h(to)e(constructs)h(on)g(and)g(b)q(et)o(w)o(een)h(the)f (corresp)q(onding)h Fi(k)q Fr([)p Fi(@)s Fr(]-mo)q(dules.)50 2179 y(A)22 b(v)m(arian)o(t)g(mo)q(dule)i(is)f(obtained)g(b)o(y)f(endo)o (wing)h(the)f(dual)h Fi(k)q Fr(-v)o(ector)f(space)h Fi(M)1515 2163 y Ff(\003)1557 2179 y Fr(with)g(a)f Fi(k)q Fr([)p Fi(@)s Fr(]-mo)q(dule)0 2233 y(structure.)27 b(Let)18 b Fi(r)h Fr(b)q(e)f(the)g (order)g(of)f Fi(L)p Fr(,)h(then)g Fi(M)23 b Fr(is)18 b(of)g(dimension)h Fi(r)g Fr(and)f(its)g(dual)g Fi(M)1569 2217 y Ff(\003)1606 2233 y Fr(=)f(Hom)1753 2240 y Fh(k)1775 2233 y Fr(\()p Fi(M)r(;)8 b(k)q Fr(\))16 b(is)0 2287 y(isomorphic)g(to)f Fi(k)311 2271 y Fh(r)330 2287 y Fr(.)20 b(No)o(w)14 b(let)i Fi(A)f Fr(b)q(e)h(the)f (companion)h(matrix)f(of)f Fi(L)h Fr(and)h(\()p Fi(b)1323 2294 y Fn(1)1342 2287 y Fi(;)8 b(:)g(:)g(:)t(;)g(b)1463 2294 y Fh(r)1481 2287 y Fr(\))15 b(b)q(e)h(the)f(canonical)i(basis)0 2341 y(of)e Fi(M)101 2325 y Ff(\003)121 2341 y Fr(.)k(The)d(latter)f(is)g(turned)h(in)o (to)f(a)g Fi(k)q Fr([)p Fi(@)s Fr(]-mo)q(dule)h(b)o(y)f(de\014ning)h(an)g(op) q(erator)e Fe(r)h Fr(on)g Fi(M)1592 2325 y Ff(\003)1627 2341 y Fr(b)o(y)g(the)g(action)617 2420 y(\()p Fe(r)p Fi(b)693 2427 y Fn(1)712 2420 y Fi(;)8 b(:)g(:)g(:)t(;)g Fe(r)p Fi(b)871 2427 y Fh(r)889 2420 y Fr(\))907 2401 y Fh(T)947 2420 y Fr(=)13 b Fe(\000)p Fi(A)1064 2401 y Fh(T)1092 2420 y Fr(\()p Fi(b)1130 2427 y Fn(1)1149 2420 y Fi(;)8 b(:)g(:)g(:)t(;)g(b)1270 2427 y Fh(r)1288 2420 y Fr(\))1306 2401 y Fh(T)0 2495 y Fr(and)17 b(letting)g Fi(@)i Fr(act)d(b)o(y)g Fe(r)p Fr(.)23 b(Th)o(us,)17 b Fe(r)p Fr(\()p Fi(am)p Fr(\))c(=)i Fi(a)p Fe(r)p Fi(m)c Fr(+)g Fi(a)1007 2479 y Ff(0)1019 2495 y Fi(m)16 b Fr(when)h Fi(a)e Fe(2)g Fi(k)i Fr(and)g Fi(m)d Fe(2)h Fi(M)5 b Fr(.)24 b(F)l(rom)16 b(this)g(Leibniz)0 2550 y(rule)g(applied)h(to)e(the)g(pro)q(duct)h Fi(y)577 2557 y Fn(1)597 2550 y Fi(b)617 2557 y Fn(1)646 2550 y Fr(+)10 b Fe(\001)e(\001)g(\001)g Fr(+)j Fi(y)822 2557 y Fh(r)841 2550 y Fi(b)861 2557 y Fh(r)892 2550 y Fr(=)i(\()p Fi(y)980 2557 y Fn(1)1000 2550 y Fi(;)8 b(:)g(:)g(:)d(;)j(y)1124 2557 y Fh(r)1142 2550 y Fr(\)\()p Fi(b)1198 2557 y Fn(1)1217 2550 y Fi(;)g(:)g(:)g(:)d(;)j(b)1339 2557 y Fh(r)1356 2550 y Fr(\))1374 2534 y Fh(T)1401 2550 y Fr(,)15 b(w)o(e)g(deriv)o(e)h(the)f (equalit)o(y)595 2629 y Fi(Y)632 2610 y Ff(0)656 2629 y Fr(=)e Fi(AY)101 b Fr(for)91 b Fi(Y)23 b Fr(=)13 b(\()p Fi(y)1148 2636 y Fn(1)1167 2629 y Fi(;)8 b(:)g(:)g(:)d(;)j(y)1291 2636 y Fh(r)1310 2629 y Fr(\))1328 2610 y Fh(T)p eop %%Page: 46 4 46 3 bop 0 22 a Fn(46)0 120 y Fr(whenev)o(er)17 b Fe(r)p Fr(\()p Fi(y)279 127 y Fn(1)298 120 y Fi(b)318 127 y Fn(1)348 120 y Fr(+)11 b Fe(\001)d(\001)g(\001)h Fr(+)j Fi(y)527 127 y 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