C\303\241lculo IILi\303\247\303\243o 31: S\303\251ries de Fourier As fun\303\247\303\265es seno e coseno desfrutam de algumas propriedades da ortogonalidade bastante \303\272teis. assume assume(n,integer): assume(m, integer): int(sin(m*x)*sin(n*x),x=-Pi..Pi);

IiIh

int(cos(m*x)*cos(n*x),x=-Pi..Pi);

IiIh

int(sin(m*x)*cos(n*x),x=-Pi..Pi);

IiIh

int(sin(n*x)*sin(n*x),x=-Pi..Pi);

SSNQaUclKnByb3RlY3RlZEc=

int(cos(n*x)*cos(n*x),x=-Pi..Pi);

SSNQaUclKnByb3RlY3RlZEc=

Utilizaremos estas rela\303\247\303\265es de ortogonalidade para desenvolvermos uma expans\303\243o em s\303\251ries para fun\303\247\303\265es, em termos de senos e cosenos. Trabalharemos no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYjUSFGJy1GIzYkLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tRjE2JVEjUGlGJy8lJ2l0YWxpY0dGP0Y6LUY3Ni1RIixGJ0Y6Rj0vRkFRJXRydWVGJ0ZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRidGUUY6LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYn, o que n\303\243o representa restri\303\247\303\243o alguma, j\303\241 que uma fun\303\247\303\243o em um intervalo [a, b] pode ser transformada, por uma transforma\303\247\303\243o linear, em uma fun\303\247\303\243o em LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobWZlbmNlZEdGJDYmLUYjNiYtSSNtaUdGJDYjUSFGJy1GIzYkLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tRjE2JVEjUGlGJy8lJ2l0YWxpY0dGP0Y6LUY3Ni1RIixGJ0Y6Rj0vRkFRJXRydWVGJ0ZCRkRGRkZIRkovRk1RJjAuMGVtRicvRlBRLDAuMzMzMzMzM2VtRidGUUY6LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYn. Para isso, definimos uma nova vari\303\241vel LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= e uma rela\303\247\303\243o linear entre LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= e x. eq1:= '(Pi-(-Pi))/(b-a)=(theta-Pi)/(x-b)';

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTFHRigvLCQqJkkjUGlHRiYiIiIsJkkiYkdGKEYySSJhR0YoISIiRjYiIiMqJiwmSSZ0aGV0YUdGKEYyRjFGNkYyLCZJInhHRihGMkY0RjZGNjcjRi4=

eq2:=theta=solve(eq1,theta);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTJHRigvSSZ0aGV0YUdGKCooSSNQaUdGJiIiIiwoSSJiR0YoRjJJImFHRihGMkkieEdGKCEiI0YyLCZGNCEiIkY1RjJGOTcjRi4=

Substituindo para x: eq3:=x=solve(eq1,x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSRlcTNHRigvSSJ4R0YoLCQqJiwqKiZJI1BpR0YmIiIiSSJhR0YoRjVGNSomRjRGNUkiYkdGKEY1RjUqJkY2RjVJJnRoZXRhR0YoRjUhIiIqJkY4RjVGOkY1RjVGNUY0RjsjRjUiIiM3I0Yu

Neste caso, a fun\303\247\303\243o \303\251 definida e peri\303\263dica em 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. O teorema da aproxima\303\247\303\243o polinomial de Weirstrass apresenta uma generaliza\303\247\303\243o para m\303\272ltiplas vari\303\241veis. Uma fun\303\247\303\243o em n vari\303\241veis, f( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn,...LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw==), pode ser aproximada, com a precis\303\243o desejada, por uma polin\303\264mio em ( LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJy1JJW1zdWJHRiQ2JS1GLzYlUSJrRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lMXN1cGVyc2NyaXB0c2hpZnRHRkhGRg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMkYnL0Y2USdub3JtYWxGJy1JJW1zdWJHRiQ2JS1GLzYlUSJrRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lMXN1cGVyc2NyaXB0c2hpZnRHRkhGRg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiM0YnL0Y2USdub3JtYWxGJy1JJW1zdWJHRiQ2JS1GLzYlUSJrRidGMkY1RjgvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy8lMXN1cGVyc2NyaXB0c2hpZnRHRkhGRg== ***LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkobXN1YnN1cEdGJDYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUtSSVtc3ViR0YkNiUtRi82JVEia0YnRjJGNUY4LyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZFRkM= ), onde LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRImlGJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJw== s\303\243o inteiros > 0.Neste estudo, nos restringimos ao caso do plano, com duas vari\303\241veis, x e y, e expressamos as vari\303\241veis em coordenadas polares, 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, 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. Restringimo-nos ainda ao c\303\255rculo unit\303\241rio, uma vez que toda fun\303\247\303\243o plana peri\303\263dica pode ser expressa como uma fun\303\247\303\243o neste c\303\255rculo. 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 = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSJmRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RicvRjhRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRlEtSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSgmdGhldGE7RicvRjVGQkY+Rj5GKw== = 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 = 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 , sendo utilizada a identidade de Euler no \303\272ltimo passo: 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 . Com isso, pode-se expandir as somas dos produtos dos termos exponenciais. A fun\303\247\303\243o a seguir realiza esta tarefa para v\303\241rios valores de a e b. E:=(a,b)->expand(sum(sum((exp(I*x)+exp(-I*x))^n*(exp(I*x)-exp(-I*x))^m, m=0..a),n=0..b)): O leitor deve estar convencido de que, para qualquer valor de a e b, a soma dos produtos resulta em uma soma de termos da forma 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| 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, 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. No entanto, a soma infinita resulta em 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 , que pode ser reescrita como: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 = 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 = 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 = 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 Todo o procedimento anterior pode ser implementado no Maple. Para fins de compara\303\247\303\243o, utilizamos novamente 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. Ex. 12.1 restart: f:=exp(sin(x));

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoLUkkZXhwR0YlNiMtSSRzaW5HRiU2I0kieEdGKDcjRi4=

A:=[seq(evalf(int(f*cos(k*x),x=-Pi..Pi)),k=0..3)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoNyYkIitDbCNcJnohIiokIiIhRjMkIStVd0ZIJikhIzVGMjcjRi4=

B:=[seq(evalf(int(f*sin(k*x),x=-Pi..Pi)),k=0..3)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJCR0YoNyYkIiIhRjAkIit5JCoqNGIkISIqRi8kISs9SylHUiIhIzU3I0Yu

assume(j>0): F:=A[1]/(2*Pi)+sum(A[j]/Pi*cos((j-1)*x)+B[j]/Pi*sin((j-1)*x),j=2..4);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJGR0YoLCokIit5ZTFtNyEiKiIiIi1JJHNpbkdGJTYjSSJ4R0YoJCIrMyM9LjgiRjEtSSRjb3NHRiU2IywkRjYiIiMkISsmUmBcciMhIzUtRjQ2IywkRjYiIiQkISsmKVxvTFchIzY3I0Yu

Note que as sequ\303\252ncias de coeficientes (B) encontram-se entre colchetes, a fim de assegurar que ser\303\243o tomadas na ordem correta. Al\303\251m disso, note que os argumentos para sen e cos na express\303\243o para a s\303\251rie de Fourier s\303\243o (j -1), e n\303\243o j. Isto ocorre porque o elementos na lista de coeficientes s\303\243o numerados, na mem\303\263ria do Maple, a partir de 1, enquanto que nas equa\303\247\303\265es estes come\303\247am por 0. O gr\303\241fico da expans\303\243o de Fourier com quatro termos \303\251 quase indistingu\303\255vel em rela\303\247\303\243o \303\240 fun\303\247\303\243o original, no intervalo (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JI21vR0YkNi1RKiZ1bWludXMwO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOi8lKXN0cmV0Y2h5R0Y6LyUqc3ltbWV0cmljR0Y6LyUobGFyZ2VvcEdGOi8lLm1vdmFibGVsaW1pdHNHRjovJSdhY2NlbnRHRjovJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0ZJLUYsNiVRI1BpRicvJSdpdGFsaWNHRjpGNS1GMjYtUSIsRidGNUY4L0Y8USV0cnVlRidGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnL0ZLUSwwLjMzMzMzMzNlbUYnRkw=). Fora deste intervalo, n\303\243o h\303\241 raz\303\243o para que se tenha uma boa aproxima\303\247\303\243o. plot([f,F],x=-Pi..Pi,color=[black,yellow],thickness=[1,4]);

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Avaliaremos agora o gr\303\241fico de uma fun\303\247\303\243o com um \303\272nico dente de serra. Ex. 12.2 restart: f:=abs(x);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoLUkkYWJzR0YmNiNJInhHRig3I0Yu

A:=[seq(evalf(int(f*cos(k*x),x=-Pi..Pi)),k=0..5)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoNygkIisvV2dwKSohIiokISIlIiIhJEY0RjQkIStXV1dXVyEjNUY1JCErKysrKztGODcjRi4=

B:=[seq(evalf(int(f*sin(k*x),x=-Pi..Pi)),k=0..5)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJCR0YoNygkIiIhRjBGL0YvRi9GL0YvNyNGLg==

Observe que, para fun\303\247\303\265es sim\303\251tricas, n\303\243o h\303\241 termos envolvendo a fun\303\247\303\243o seno. assume(j>0): F:=A[1]/(2*Pi)+sum(A[j]/Pi*cos((j-1)*x),j=2..6);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJGR0YoLCokIitGanpxOiEiKiIiIi1JJGNvc0dGJTYjSSJ4R0YoJCErWCZSS0YiRjEtRjQ2IywkRjYiIiQkISswMXI5OSEjNS1GNDYjLCRGNiIiJiQhK3kiZUg0JiEjNjcjRi4=

A expans\303\243o de Fourier com apenas quatro termos apresenta uma boa aproxima\303\247\303\243o. plot([f,F],x=-Pi..Pi,color=[red,black]);

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Ex. 12.3 Consideremos agora um \303\272nico ciclo de uma onda quadrada. Esta fun\303\247\303\243o \303\251 igual a 1 no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JkYrLUYjNiQtSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjwvJSlzdHJldGNoeUdGPC8lKnN5bW1ldHJpY0dGPC8lKGxhcmdlb3BHRjwvJS5tb3ZhYmxlbGltaXRzR0Y8LyUnYWNjZW50R0Y8LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSy1GLDYlUSNQaUYnLyUnaXRhbGljR0Y8RjctRjQ2LVEjLi5GJ0Y3RjpGPUY/RkFGQ0ZFRkdGSS9GTVEmMC4wZW1GJy1JI21uR0YkNiRRIjBGJ0Y3Ris=, e igual a -1 no intervalo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRIjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUSMuLkYnRjUvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj4vJSlzdHJldGNoeUdGPi8lKnN5bW1ldHJpY0dGPi8lKGxhcmdlb3BHRj4vJS5tb3ZhYmxlbGltaXRzR0Y+LyUnYWNjZW50R0Y+LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictRiw2JVEjUGlGJy8lJ2l0YWxpY0dGPkY1Ris=. Encontrar sua representa\303\247\303\243o de Fourier. restart: f:=piecewise(x<0,1,x<-Pi,0,x>0,-1,x>Pi,0);

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoLUkqUElFQ0VXSVNFR0YlNiY3JCIiIjJJInhHRigiIiE3JEY1MkY0LCRJI1BpR0YmISIiNyRGOjJGNUY0NyRGNTJGOUY0NyMtSSpwaWVjZXdpc2VHRiY2KkYzRjJGN0Y1RjxGOkY+RjU=

A:=[seq(evalf(int(f*cos(k*x),x=-Pi..Pi)),k=0..20)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJBR0YoNzckIiIhRjBGL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvRi9GL0YvNyNGLg==

No caso de fun\303\247\303\265es anti-sim\303\251tricas, s\303\243o os termos em coseno que desaparecem. B:=[seq(evalf(int(f*sin(k*x),x=-Pi..Pi)),k=0..20)];

LV9JLFR5cGVzZXR0aW5nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJCR0YoNzckIiIhRjAkISIlRjBGLyQhK0xMTEw4ISIqRi8kISsrKysrISkhIzVGLyQhKzlkRzlkRjhGLyQhK1dXV1dXRjhGLyQhK09PT09PRjhGLyQhK3hJI3AyJEY4Ri8kIStubW1tRUY4Ri8kISt3NiVITiNGOEYvJCErZUpFMEBGOEYvNyNGLg==

assume(j>0): F:=A[1]/(2*Pi)+sum(A[j]/Pi*cos((j-1)*x)+B[j]/Pi*sin((j-1)*x),j=2..21);

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plot([f,F],x=-Pi..Pi,color=[red,black]);

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Foram ilustradas v\303\241rias fun\303\247\303\265es definidas no intervalo 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. Se estas fossem definidas em algum outro intervalo, a mudan\303\247a de vari\303\241vel indicada acima resultaria em uma modifica\303\247\303\243o na expans\303\243o de Fourier: 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 O per\303\255odo das fun\303\247\303\265es trigonom\303\251tricas altera-se de acordo com o fator de aumento devido ao mapeamento de um intervalo arbitr\303\241rio de tamanho L em um intervalo de tamanho LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1JI21uR0YkNiRRIjJGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYtUTEmSW52aXNpYmxlVGltZXM7RidGNS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPi8lKXN0cmV0Y2h5R0Y+LyUqc3ltbWV0cmljR0Y+LyUobGFyZ2VvcEdGPi8lLm1vdmFibGVsaW1pdHNHRj4vJSdhY2NlbnRHRj4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZNLUYsNiVRI1BpRicvJSdpdGFsaWNHRj5GNUYr. Neste caso, os limites de integra\303\247\303\243o tornam-se as extremidades do novo intervalo. Curiosamente, o Maple n\303\243o possui uma fun\303\247\303\243o que gera uma s\303\251rie de Fourier correspondente \303\240 determinada fun\303\247\303\243o. Aplica\303\247\303\265es 1. Encontrar a expans\303\243o de Fourier, com 4 e 8 termos, para LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUSdhcmNzZW5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUTAmQXBwbHlGdW5jdGlvbjtGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkIvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGUS1JKG1mZW5jZWRHRiQ2JC1GIzYlRistRiM2JS1GLDYlUSRzZW5GJ0Y0RjdGOi1GVTYkLUYjNiMtRiw2JVEieEYnRjRGN0Y+RitGPkYr, sobre o intervalo 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. Tra\303\247ar o gr\303\241fico da fun\303\247\303\243o e de suas expan\303\247\303\265es no mesmo conjunto de eixos. 2. Expandir 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, 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 como uma expans\303\243o de Fourier. Quantos termos s\303\243o necess\303\241rios para que se tenha uma aproxima\303\247\303\243o "razo\303\241vel"? Como se pode definir "razo\303\241vel"?