plancheCentrale2.html

EXERCICE 1

> with(LinearAlgebra):

> S := <<-1,0>|<0,1>>;

> T := (1/sqrt(2))*<<-1,1>|<1,1>>;

> R := (S.T);

> seq(R^k,k=0..8);

> S.R;

> Phi := M -> S.M;

> liste := table([seq(R^k,k=0..7),seq(R^k.S,k=0..7)]);

>    sigma := proc(k)
local j;
for j from 1 to 16 do
    if Equal(Phi(liste[k]),liste[j])
    then RETURN(j);
    fi;
od;
end:

> sigma(1);

> sigma := table([seq(sigma(k),k=1..16)]);

> signature := product(product((sigma[j] - sigma[i])/(j-i),j=i+1..16),i=1..16);

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EXERCICE 2
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> restart;

> u := n -> evalf(product(1 + I/k^2, k = 1..n));

> v := n -> evalf(product(1 + 2*I/k, k = 1 .. n));

> P := (w,N) -> plot([seq([Re(w(j)),Im(w(j))],j = 1 .. N)]):

> P(v,500);

> P(u,200);

> l := limit(u(n),n=infinity);

> abs(l);

> argument(l);

> abs(u(6));

> argument(u(1000));

>

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EXERCICE 3
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> P := (n,x) -> product((1 + x/(2*k)) / (1 + x/(2*k - 1)), k = 1..n);

> assume(n,posint);simplify(limit(P(n,x),x=infinity));

> plot([seq(P(n,x),n = 1..30)],x=0..20,thickness=[3,1$9]);

> p := x -> simplify(limit(P(n,x),n=infinity));

> assume(j,posint);

> simplify(p(2*j));

> seq(p(2*j),j=1..10);

> pj := m -> ((2*m)!/((m!)^2*2^(2*m)));

> seq(pj(j),j=1..10);

> series(pj(j),j=infinity,2);

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EXERCICE 4
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> # question a

> restart;

> M1 := s -> [arcsinh(s), sqrt(1+s^2)];

> G := (M,s) -> map(z -> (1/s)*int(z,u = 0..s), M(u));

>

> Gamma := M -> plot([seq(M(0.1*s),s = 0..700)],color=blue):

> Delta := M -> plot([seq(G(M,0.1*s),s = 0..1500)],color=red):

> P := M -> plots[display](Gamma(M),Delta(M));

> P(M1);

> M2 := s -> [cos(s),sin(s)]:

> P(M2);

> G(M2,s);

> plots[display](Gamma(M2),Delta(M2),seq(plot([M2(s),G(M2,s)],color=wheat),s in [Pi/2,3*Pi/4,Pi]));

>

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EXERCICE 5
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> A := (X - 1)*(X - 2)*(X - 3):

> B := X^3:

> f := P -> rem(B*P,A,X);

> P := a*X^2 + b*X + c;

> solve([seq(coeff(f(P),X,k) = 0,k = 0..2)],[a,b,c]);

> Q := collect(f(P) - lambda*P,X);

> solve([seq(coeff(Q,X,k) = 0,k=0..2)],[a,b,c,lambda]);

> with(LinearAlgebra):

> mat := f -> Transpose(Matrix(3,3,[seq(seq(coeff(f(E),X,k),k=0..2),E in [1,X,X^2])])):

> # Ou bien : mat := f -> <<seq(coeff(f(1),X,k),k=0..2)>|<seq(coeff(f(X),X,k),k=0..2)>|<seq(coeff(f(X^2),X,k),k=0..2)>>:

> # ou bien : mat:= f -> Matrix(3,3,[seq(PolynomialTools[CoefficientVector](f(E),X), E in [1,X,X^2])]);

> mat(f);

> Eigenvectors(mat(f));

> ?lagrange;

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EXERCICE 6
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> restart;

> A := Matrix(6,6,(i,j) -> if j-i = 1 or (i,j) = (1,6) then 1 else 0 fi);

> M := Matrix(6,6,(i,j) -> m[i,j]);

> B := A.M - 2*M.A:

> S := solve([seq(seq(B[i,j], j = 1..6),i=1..6)],[seq(seq(m[i,j], j = 1..6),i=1..6)]);

> ?assign;

> assign(S);

> 4*M;

>

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EXERCICE 7
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> i1 := int(t^n*(1 - t)^m,t = 0..1);