Uebung 131)a)restart;
f1:=x+1;
f2:=x^3+3*x;
f3:=x^5+5*x^3;
eq:=c0*f1+c1*f2+c2*f3;
eq:=sort(expand(eq));
cc0:=coeff(eq, x);
cc1:=coeff(eq, x^3);
cc2:=coeff(eq, x^5);
solve({cc0,cc1,cc2},{c0,c1,c2});b)restart;
f1:=1;
f2:=x+1;
f3:=2*x+3;
eq:=c0*f1+c1*f2+c2*f3;
eq:=sort(expand(eq));
cc0:=coeff(eq, x);
cc1:=coeff(eq, x);
cc2:=coeff(eq, x);
solve({cc0,cc1,cc2},{c0,c1,c2});2)a)legendre:=proc(n,x)
if n=0 then return 1:
elif n=1 then return x:
else return ( 1/n*(2*n-1)*x*legendre(n-1,x)-(n-1)*legendre(n-2,x) ):
end if:
end proc:legendre(5,x);
simplify(%);leg_seq:=seq( simplify(legendre(n,x)),n=0..5);orth:=(i,j) -> int(legendre(i,x)*legendre(j,x), x=-1..1);
orth_seq:=seq([seq(orth(i,j), i=0..5)], j=0..5);3)c)leg_reihe:=proc(f,ordnung,a,b)
local i, pl,p:
pl:=[]:
p:=(x)->sum( (2*k+1)/2*int(f(x)*legendre(ordnung,x),x=a..b) * legendre(k,x) ,'k'=0..ordnung):
p:=evalf(simplify(p)):
for i from 0 to ordnung do
end do:
print(p);
end proc:f:=x->sin(Pi*x):
leg_reihe(f,5,-1,1);p:=(x)->sum( (2*k+1)/2*int(f(x)*legendre(ordnung,x),x=a..b) * legendre(k,x) ,'k'=0..ordnung);