{VERSION 4 0 "IBM INTEL LINUX22" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 0 0 255 1 0 2 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 2 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 } {PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 2 2 2 2 2 2 1 1 1 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 36 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 18 "" 0 "" {TEXT 256 22 "Variable complexe (06)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "with(plots): with(pldx): with(linalg): kernelopt s(ASSERT=true):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "Thales" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "n:=7: lip:= seq([k,0],k=0..n );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "liq:= seq([k*1.4,k*0. 2],k=0..n);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "pl1:=pointpl ot([liq[n+1],lip[1],lip[n+1]], style=line, linestyle=24):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "pl2:=map(pointplot, [seq([lip[k],li q[k]],k=1..n+1)], style=line, linestyle=16):;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "pl3:= textplot([[-0.3,-0.3,\"A\"],[7,-0.3,\"B\"] ,[10.2,1.7,\"C\"]]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "dis playf(pl1,pl2, pl3, axes=none);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([x/2, x/2+1],x=-5..5,color=black, linestyle=24);" }}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 29 "Th\351or\350me de l'angle au cent re" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cp:= linestyle=24, col or=black;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "[-1,0],[3/5,4 /5]; map( (proc(x,y) -y+a*x+b end)@op, \{%\});\nsubs(solve(%), a*x+b); pl3:= plot(%, x=-1.2..1, cp):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pl1:= plot([cos(t),sin(t),t=0..2*Pi], cp): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "pl2:= pointplot([0,0,1,4/3], style=line, cp),\npointplot([1.1,0,-1.1,0,0,0,0,1.1], style=line, cp):\npl22:= po intplot([0,0,1.+3/5,4./5,3./5,4./5], style=line, cp):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "pl4:= textplot([[-1.1,0.1,\"S\"],[ .57,.95,\"M\"],[-.1,0.6,\"P\"],[.9,-0.1,\"A\"],[0,-0.1,\"O\"]]):\npl44 := textplot([[1.6,.95,\"N\"],[1.,0.42,\"v\"]]):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 90 "displayf(pl1, pl2, pl3, pl4, axes=none, scalin g=constrained,\nview=[-1.2..1.7, -0.2..1.1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "displayf(pl1, pl2, pl22, pl3, pl4, pl44, axes=n one, scaling=constrained,\nview=[-0.2..1.7, -0.2..1.1]);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Une \351quation du second degr\351" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "(1+I)*(z-2-I)*(z+3-2*I); col lect(expand(%), z);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 6 "Cercle" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 85 "pts:= [2-3*I,-1+2*I,1+I,x+I*y];\nmatrix(map([abs^2,Re,Im,1],pt s)): ma:= map(evalc,%);;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "dev:= add((-1)^j*ma[4,j]*fu(submatrix(ma,[1,2,3],subs(j=NULL,[1,2,3,4 ])) ), j=1..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "(eval@su bs)(fu=(z->z), dev);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "eqc := (eval@eval@subs)(fu=det, dev);\neqcc:= xcs(%/coeff(%,x^2),x,y); " } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "rc:=-remove(has,eqcc,\{x,y \}); cc:= x-op(1,select(has,eqcc,x))+I*(y-op(1,select(has,eqcc,y)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "seq(abs(k-cc)^2, k=pts); \+ " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "birapport" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "((d-b)/(d-a)): %/subs(d=c,%): beta:= unappl y(%,a,b,c,d);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "beta(1,-1, I,-I);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "birap:= (normal@e valc@[Re,Im]@beta@op)(pts);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "numer(birap[1]); eqcd:= xcs(%/coeff(%,x^2),x,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "rd:=-remove(has,eqcd,\{x,y\}); cd:= x-op( 1,select(has,eqcd,x))+I*(y-op(1,select(has,eqcd,y)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "pl1:= complexplot(cc+(cos(t)+I*sin (t))*sqrt(rc), t=0..2*Pi, linestyle=16):\npl2:= complexplot(cd+(cos(t) +I*sin(t))*sqrt(rd), t=0..2*Pi, linestyle=16):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "ro:=0.2: nu:=8; pate:= proc(a,b);\ntry \npol ygonplot([seq([a+cos(t*2*Pi/nu)*ro,b+sin(t*2*Pi/nu)*ro],t=0..nu)], col or=blue)\ncatch: end try;\nend:;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "pl3:= map(pate@op@[Re,Im], pts): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "displayf(pl1,pl2, pl3, scaling=constrained, tickmarks=[[-4,7],[-4,4]]);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "algorithme de newton" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "new := unapply(``(1/2)*(z+alpha^ 2/z),z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "h:=unapply((z-alpha)/(z+alpha),z); hi:= unapply(solve (h(z)=zeta,z),zeta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "(no rmal@yjoli@h@new@hi)(zeta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "nev:= proc(z) option remember; subs(alpha^2=2+I, (yjoli@new)(z)); \+ end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "Vector([seq((nev@@k )(1.), k=0..7)]), Vector([seq((nev@@k)(I*1.), k=0..7)]);;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "subs(alpha= (nev@@7)(1.), [h(1), h( I)]): map(abs,%);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Conformal" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "conformalin(z, -3-3*I..3+ 3*I, -3-3*I..3+3*I, scaling=constrained, color1=blue, color2= magenta, grid=[13,13]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "conform alin((z-1)/(z+1), -4-3*I..2+3*I, -2-3*I..4+3*I, scaling=constrained, \+ \ncolor1=blue, color2= magenta, grid=[13,13], numxy=[60,60], axes=boxe d);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "Stereographique" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ster:= a=2*x/(1+x^2+y^2), b= 2*y/(1+x^2+y^2),c=(1-x^2-y^2)/(1+x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "fster:= unapply(subs(ster,[a,b,c]),x,y);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sph:= a^2+b^2+c^2-1;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "(normal@subs)(ster, sph);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "S:= [0,0,-1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "mm:= matrix([[a,b,c],[x,y,0],S]):\n subs(ster, %); ASSERT(det(%)=0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "cir:= x^2+y^2-2*alpha*x-2*beta*y+gamma=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "xcs(lhs(cir),x,y):select(has,%,\{x, y\})=-remove(has,%,\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eli:= eliminate(\{ster\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eli[2]: simplify(%,\{a^2+b^2+c^2-1\}): (collect)(fact or(%),y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rets:= op(eli[ 1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(rets, eval(mm) ); det(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "(numer@subs) (rets, lhs(cir))-sph;\nmap(collect, factor(%), [a,b,c]);\nrel:= select (has,%,alpha);\nselectremove(has,%,\{a,b,c\});\nconj:= collect(-%[1]/% [2]=1, \{a,b,c\}, normal);\nprel:= unapply(rel,a,b,c);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "Q:= map2(coeff,lhs(conj),[a,b,c]); \nsubs(seq([a,b,c][j]=%[j],j=1..3), sph);\ncollect(%, [alpha,beta], no rmal):\nfactor(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "CC:= \+ [seq(k*Q[j],j=1..3)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "(prel@op)( CC): (op@solve)(%,\{k\});\nCC:= (factor@subs)(%,CC);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "(normal@prel@op)(CC);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Omega:=[0$3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "IMC:= subs(ster, x=alpha,y=beta,[a,b,c]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "matrix([IMC,Q,S]); det(%); " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "fig:= expr-> subs(rho=sqrt( alpha^2+beta^2-gamma), alpha= 1/3, beta=2/3, gamma=1/9, expr);\nfig(co nj); fig(Q); fig(CC); simplify(fig(rho));" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 174 "pl1:= spacecurve([cos(t),sin(t),0,t=0..2*Pi], colo r=red),\nspacecurve([tan(t),0,0,t=-Pi/2..Pi/2], color=red, numpoints=1 00),\nspacecurve([0,tan(t),0,t=-Pi/2..Pi/2], color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "pl2:= spacecurve( fig([alpha+rho*co s(t),beta+rho*sin(t),0,t=0..2*Pi]), color=blue, linestyle=24):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "ro:=0.04: \npl3:= spacecurve ([fig(alpha)+ro*cos(t),fig(beta)+ro*sin(t),0,t=0..2*Pi], color=blue): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "di:= display(pl1, pl2, pl3, axes=boxed, scaling=constrained, orientation=[-90,0], view=[-1.5 ..2.5,-1.5..2.5,-1..1],\ntickmarks=[[0,1],[0,1],[]]): %;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dio:= plottools[transform]((x,y,z)- >fster(x,y))(di):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "ql1:= \+ pointplot3d(fig([S,Q,CC,IMC, Omega]), color=black, symbol=circle):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "# , tickmarks=[[-1,1],[-1,1 ],[-1,1]]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "display(dio, \+ ql1, labels=[a,b,c], orientation=[33,74], scaling=constrained, tickmar ks=[[-1,1],[-1,1],[-1,1]], axes=framed);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 "autre chose" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 " (fig@lhs)(cir); solve(subs(y=0,%),\{x\});" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "beta:= 'beta';" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "mh:= < |>; mg:= <<1,1>|<-alpha,-beta>>;\nmig:= <|<-alpha,-1 >>; mv:=<<``|``>,<``|``>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "mg . mig;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "mtmp:= map( collect, mg.mh.mig, [alpha,beta], distributed);" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 41 "res:= <, >;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "mkt:= proc(a) local n,m,j,k ; n:=rowdim(res); m:=coldim(res);\nprint(n,m); for j from 1 to n do fo r k from 1 to m do\nprint(a[j,k]); od;od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "z= (a*z+b)/(c*z+d); eqz:= unapply((numer@(lhs-rhs))(%),z);\nexpand(c*(z-a lpha)*(z-beta)); rels:= coeffs(eqz(z)-%,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "qu:= mtmp[1,1]/mtmp[2,2]; simplify(qu, \{rels\}); \+ factor(%); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "-b=-b-eqz(al pha), b=b+eqz(beta); (factor@subs)(%, qu);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 85 "hh:= unapply((2*z-I)/(z-1),z); solve(z=hh(z));\n[ev alf(%)]; map(D(hh), %);\nmap(abs,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "expand((z-1)*(z-I)); c=coeff(%,z,2), b=-coeff(%,z,0) , a=d-coeff(%,z,1);\nhh:= unapply(subs(%, d=1, (a*z+b)/(c*z+d)), z); p f:= solve(hh(z)=z,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ma p(D(hh),[pf]); map(abs,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve((hh@@3)(z)=-1); (hh@@4)(%);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Produits scalaires" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "Ps:= (f,g)-> Int(f(t)*g(t),t=0..1); ps:= (f,g)-> int(f(t)*g(t) ,t=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x:= t->t;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ps(1,1); ps(x,x);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "e0:=1; e1:= x-1/2; e2:= x^2+ (t->a)*x+(t->b); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "ps(e0, e2), ps(e1,e2); solve(\{%\}); e2:= subs(%, eval(e2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ps(e0,e0); ps(e1,e1); ps(e2,e2);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:= x^2+x+1; ps(f,f);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "cf:= proc(f,e); ps(f,e)/ps(e ,e); end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "seq(cf(f,e||k) , k=0..2); add(cf(f,e||k)*``(e||k), k=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "add(cf(f,e||k)^2*``(ps(e||k, e||k)), k=0..2): %= expand(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f:= x^3; ps(f,f);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "cf:= proc(f,e); ps(f,e)/ps(e,e); end;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "seq(cf(f,e||k), k=0..2); a dd(cf(f,e||k)*``(e||k), k=0..2);\ne3:= x^3-expand(%); seq(ps(e3, e||j) ,j=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "add(cf(f,e||k) ^2*``(ps(e||k, e||k)), k=0..2): %=expand(%);\nps(f,f)-rhs(%); ps(e3,e3 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "mp:= matrix(3,3,[1,-1/2,1/6,0,1,-1,0,0,1]);\nmg:= matrix(3,3,(j,k)->1 /(j+k-1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalm(transpo se(mp) &* mg &* mp);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 8 "DS final" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 28 "Une \+ \351quation du second degr\351" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "aa:= 1; xx:= 1+I; yy:= xx*(3-I)^2;\ncollect(aa*(z-xx)*(z-yy), z ); \nsubs(coeff(%,z)=a,%): eq:= unapply(%,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "subs(z=3+2*I, eq(a)); solve(%,a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "discrim(eq(a),z); solve(%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "eq(-15-3*I); solve(%); discr im(%%,z); abs(%); sqrt(%%);" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "pf:= I*sqrt(3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "( z-pf)/(z+pf)*exp(2*I*Pi/6)=(zeta-pf)/(zeta+pf):\n(solve)(%,zeta): f:= \+ unapply(%,z); f(solve(denom(f(z)))):= infinity;\nf(infinity):= limit(f (z),z=infinity);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalc(f (I*k));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "phi:= unapply((z +pf)/(z-pf),z); psi:= unapply(solve(phi(z)=zeta,z),zeta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "h2m:= proc(h); [numer,denom](h(z)); map(coeff,%,z),map(coeff,%,z,0);\nMatrix(matrice([%])) end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "mf:=h2m(f); mphi:=h2m(phi); \+ mpsi:=h2m(psi); mv:= Matrix(2,2,(j,k)->``);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "<, >;\n%[-2,-2]/%[-1,-1]: %= evalc(%);" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "(normal@phi@f@psi)(zet a); collect(%,zeta,evalc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 29 "liz:= seq((f@@k)(I),k=0..6); " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 80 "pl2:= complexplot([I*sqrt(3),%], style=point, \+ scaling=constrained, color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pl1:= plot([cos(t),sin(t)+2,t=0..2*Pi]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "displayf(pl1,pl2, tickmarks=[[-1,+1 ],[1,3]],view=[-1.1..+1.1,0..3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "seq((f@@k)(I),k=0..3); ((%[3]-%[2])/(%[3]-%[1])) / (( %[4]-%[2])/(%[4]-%[1]));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "nn:= 1: [x+I*y,pf,-pf,liz[nn]]; (evalc@map)([abs^2,Re,Im,1],%):\n matrice(%); det(%);\nsubs(x=Re(liz[nn+3]),y=Im(liz[nn+3]),%);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 158 "nn:= 2: [x+I*y,pf,-pf,liz[nn]]; (evalc@map)([abs^2 ,Re,Im,1],%):\nmatrice(%); det(%): coeff(%,x,2): ``(%)*normal(%%/%);\n subs(x=Re(liz[nn+3]),y=Im(liz[nn+3]),%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "nn:= 3: [x+I*y,pf,-pf,liz[nn]]; (evalc@map)([abs^2,R e,Im,1],%):\nmatrice(%); det(%): coeff(%,x,2): ``(%)*normal(%%/%);\nsu bs(x=Re(liz[nn+3]),y=Im(liz[nn+3]),%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "Sum(z^k/k/(k+1),k=1..infinity)=add(z^k/k/(k+1),k=1. .6);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "fu:= collect(sum(z^ k/k/(k+1),k=1..infinity), ln, expand@normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "gu:= (fu+subs(z=-z, fu))/2; series(%,z,10); " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "collect(expand(2*gu), 1/z, u->combine(u,ln,symbolic))/2: subs(1-z=(1-z^2)/(1+z),%):\napplyop(sim plify,3,%); series(%,z); \n(subs)(z=sqrt(y),%%); convert(%,list); map( series,%,y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "series(arct anh(z),z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "Sum(z^k/(2*k)/(2*k+1),k=1..infinity); value(%); serie s(%,z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "# gu:= collect(sum(z^k/k/(k-1),k=2..infinity), ln, \+ normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "limit(fu,z=1); \+ limit(gu,z=1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "collect(diff(fu,z), ln, normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "def:= f(z)=Sum(z^k/k/(k+1),k=1..infinity) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "op([2,1], def)/z^k; co nvert(%, parfrac,k);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "su: = sum(z^k/k, k=1..infinity); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "eqd:= Diff(f(z)-f(z)/z+1,z)= (f(z)/z-1)/z; collect((eval@subs)(f (z)=su, value(%)), ln, normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "collect(z*diff(fu,z),ln, normal); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "1/k/(k+1)/(k+2); convert(%, parfrac,k);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "1/k/(k+1): %-subs(k=k+1,%); \+ normal(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "Sum(z^k/k/(k+ 1)/(k+2), k=1..infinity); collect(value(%), ln, normal);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "mk_ma:= proc(uu); matrix(4,4,(j,k)-> (min(j+uu,k)+min (j,k+uu)-1)); end; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "mo:= mk_ma(0); xjoli(evalm(mk_ma(1/2)-%)); ms:= op(2,%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "evalm(mo+t*ms); collect(charpoly(%,z),z); \nfsolve(subs(z=0,%)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 " mk_ma(.9); fsolve(charpoly(%,z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "ma:= mk_ma(1/2); ve:= [x,y,z,t];\nps:= (a,b)-> evalm( a &* ma &* b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ps (ve,ve ); xcs(%,op(ve));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "for k \+ to 4 do e||k:=[0$(k-1),1,0$(4-k)]; od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "f||1:=e||1: for k from 2 to 4 do \n f||k:=e||k-add(f| |j*ps(e||k,f||j)/ps(f||j,f||j), j=1..k-1);\nod;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "ps(e3,f1); ps(e3,f2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "pa s:= matrice([f1,f2,f3,f4]); sap:= inverse(%); evalm(transpose(pas) &* \+ ma &* pas);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "w:= [1,1,1,1 ]; evalm(sap &* w); add(%[k]^2*ps(f||k,f||k),k=1..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ps(w, w);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "79" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }