> > # Tehtava 6 > assume(n,integer): -------------------------------------------------------------------------------- > f:=100*(Heaviside(x+4)-Heaviside(x+2)); f := 100 Heaviside(x + 4) - 100 Heaviside(x + 2) -------------------------------------------------------------------------------- > a[0]:=(1/(2*4))*int(f,x=-4..4); a[0] := 25 -------------------------------------------------------------------------------- > a[n]:=(1/4)*int(f*cos(n*Pi*x/4),x=-4..4); a[n~] := 4 / | 1/4 | (100 Heaviside(x + 4) - 100 Heaviside(x + 2)) cos(1/4 n~ Pi x) dx | / -4 -------------------------------------------------------------------------------- > # Ei onnistunut nain. (Ainakaan talla Maple -versiolla) > # Taytyy kayttaa kiertotieta > a[n]:=(1/4)*int(100*cos(n*Pi*x/4),x=-4..-2); sin(1/2 n~ Pi) sin(n~ Pi) a[n~] := - 100 -------------- + 100 ---------- n~ Pi n~ Pi -------------------------------------------------------------------------------- > a[n]:=simplify("); sin(1/2 n~ Pi) - sin(n~ Pi) a[n~] := - 100 --------------------------- n~ Pi -------------------------------------------------------------------------------- > # Maple ei nakojaan automaattisesti tieda, etta sin(n*Pi) = 0 > b[n]:=simplify((1/4)*int(100*sin(n*Pi*x/4),x=-4..-2)); cos(1/2 n~ Pi) - cos(n~ Pi) b[n~] := - 100 --------------------------- n~ Pi -------------------------------------------------------------------------------- > # Maple ei liioin tieda suoraan, etta cos(n*Pi) = (-1)^n > sarja:=a[0]+sum(a[n]*cos(n*Pi*x/4)+b[n]*sin(n*Pi*x/4),n=1..infinity); infinity ----- \ (sin(1/2 n~ Pi) - sin(n~ Pi)) cos(1/4 n~ Pi x) sarja := 25 + ( ) (- 100 ---------------------------------------------- / n~ Pi ----- n~ = 1 (cos(1/2 n~ Pi) - cos(n~ Pi)) sin(1/4 n~ Pi x) - 100 ----------------------------------------------)) n~ Pi -------------------------------------------------------------------------------- > sarja:=simplify("); infinity ----- \ sarja := 25 (Pi - 4 ( ) (cos(1/4 n~ Pi x) sin(1/2 n~ Pi) / ----- n~ = 1 - cos(1/4 n~ Pi x) sin(n~ Pi) + sin(1/4 n~ Pi x) cos(1/2 n~ Pi) - sin(1/4 n~ Pi x) cos(n~ Pi))/n~))/Pi -------------------------------------------------------------------------------- > # Kokeillaan x:n arvolla 5 > subs(x=5,sarja); infinity ----- \ 25 (Pi - 4 ( ) (cos(5/4 n~ Pi) sin(1/2 n~ Pi) - cos(5/4 n~ Pi) sin(n~ Pi) / ----- n~ = 1 + sin(5/4 n~ Pi) cos(1/2 n~ Pi) - sin(5/4 n~ Pi) cos(n~ Pi))/n~))/Pi -------------------------------------------------------------------------------- > simplify("); infinity ----- \ 25 (Pi - 4 ( ) (cos(5/4 n~ Pi) sin(1/2 n~ Pi) - cos(5/4 n~ Pi) sin(n~ Pi) / ----- n~ = 1 + sin(5/4 n~ Pi) cos(1/2 n~ Pi) - sin(5/4 n~ Pi) cos(n~ Pi))/n~))/Pi -------------------------------------------------------------------------------- > # Ei siis osannut sieventaa, mutta tuloksen pitaisi kaiketi olla = 100 > osasum:=seq(a[0]+sum(a[n]*cos(n*Pi*x/4)+b[n]*sin(n*Pi*x/4),n=1..N),N=1..14): -------------------------------------------------------------------------------- > with(plots): -------------------------------------------------------------------------------- > plot({osasum},x=-12..12); -------------------------------------------------------------------------------- >