restart;
#Section 4.7 number 3b
N:=4;

#Create the Legendre Polynomials
P[0]:=1;
for n from 1 to N
do
    P[n]:=1/(n!*2^n)*diff((x^2-1)^n,x$n);
end;





dp:=(f,g)->int(f*g,x=-1..1);
nm:=f->sqrt(dp(f,f));


f:=log(1+x^2);

for n from 0 to N
do
    beta[n]:=dp(f,P[n])/dp(P[n],P[n]);
end;





p:=collect(sum(beta[j]*P[j],j=0..N),x);

plot({p,f},x=-1..1);

#plot of the errors on [-1,1]
plot(f-p,x=-1..1);