Project 2, Application 9.3 unit := (t,a,b) -> Heaviside(t-a) - Heaviside(t-b);

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Define the Fourier coefficients for even (a(n)) or odd (b(n)) extensions and the corresponding Fourier series a := (f,n) -> (2/Pi)*int(f(t)*cos(n*t),t=0..Pi); b := (f,n) -> (2/Pi)*int(f(t)*sin(n*t),t=0..Pi); cossum := (f,N,t) -> a(f,0)/2 + sum(a(f,n)*cos(n*t),n=1..N); sinsum := (f,N,t) -> sum(b(f,n)*sin(n*t),n=1..N);

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f1 := t -> unit(t,0,Pi/2) - unit(t,Pi/2,Pi);

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plot(f1(t),t=0..Pi);

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cossum(f1,50,t);

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The pattern here is:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= f1bis := (t,N) -> (4/Pi)*sum((-1)^k * cos((2*k+1)*t)/(2*k+1),k=0..N);

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Just checking: simplify(f1bis(t,24)-cossum(f1,50,t));

IiIh

plot({cossum(f1,50,t),f1(t)},t=0..Pi,color=black);

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f2 := t -> t*unit(t,0,Pi/2) + (Pi-t)*unit(t,Pi/2,Pi);

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plot(f2(t),t=0..Pi);

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f3 := t -> (Pi/2-t); plot(f3(t),t=0..Pi);

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCZJI1BpRyUqcHJvdGVjdGVkRyMiIiIiIiM5JCEiIkYlRiVGJQ==

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

The EVEN extension (fig 9.3.10) plot(cossum(f3,50,t),t=-2*Pi..3*Pi); cossum(f3,50,t);

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

Here the pattern repeats skips every other even number in the cos summation (it repeats every 4) f3e := (t,N) -> (4/Pi)*sum(cos((2*k+1)*t)/((2*k+1)^2),k=0..N);

Zio2JEkidEc2IkkiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJkkjUGlHJSpwcm90ZWN0ZWRHISIiLUkkc3VtRzYkRi1JKF9zeXNsaWJHRiU2JComLUkkY29zR0YxNiMqJiwmSSJrR0YlIiIjIiIiRjxGPDkkRjxGPEY5ISIjL0Y6OyIiITklRjwiIiVGJUYlRiU=

Checking simplify(f3e(t,24)-cossum(f3,50,t));

IiIh

The ODD extension (fig 9.2.4) plot(sinsum(f2,50,t),t=-Pi..Pi); sinsum(f2,50,t);

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

Here the pattern is alternating signs for all odd numbers in teh sines summation. f2o := (t,N) -> (4/Pi)*sum((-1)^k*sin((2*k+1)*t)/(2*k+1)^2,k=0..N);

Zio2JEkidEc2IkkiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKCIiJSIiIkkjUGlHJSpwcm90ZWN0ZWRHISIiLUkkc3VtRzYkRi9JKF9zeXNsaWJHRiU2JCooKUYwSSJrR0YlRi0tSSRzaW5HRjM2IyomLCYqJiIiI0YtRjhGLUYtRi1GLUYtRiRGLUYtKUY9Rj9GMC9GODsiIiFGJkYtRi1GJUYlRiU=

simplify(f2o(t,24)-sinsum(f2,50,t));

IiIh

f3 := t-> unit(t,0,Pi/3)*t + unit(t,Pi/3,2*Pi/3)*(Pi/3) + (Pi-t)*unit(t,2*Pi/3,Pi);

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJi1JJXVuaXRHRiU2JUYkIiIhLCQqJiMiIiIiIiRGMkkjUGlHJSpwcm90ZWN0ZWRHRjJGMkYyRiRGMkYyKihGMUYyLUYsNiVGJEYvLCQqJiMiIiNGM0YyRjRGMkYyRjJGNEYyRjIqJiwmRjRGMkYkISIiRjItRiw2JUYkRjlGNEYyRjJGJUYlRiU=

plot(f3(t),t=0..Pi);

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

plot(sinsum(f3,48,t),t=-Pi..Pi); sinsum(f3,48,t);

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

Here the pattern repeats every 6 terms f3o := (t,N) -> (2*sqrt(3)/Pi)*sum(sin((6*k+1)*t)/(6*k+1)^2 - sin((6*k+5)*t)/(6*k+5)^2,k=0..N);

Zio2JEkidEc2IkkiTkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqKiIiIyIiIi1JJXNxcnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyIiJEYtSSNQaUdGMSEiIi1JJHN1bUdGMDYkLCYqJi1JJHNpbkdGMDYjKiYsJiomIiInRi1JImtHRiVGLUYtRi1GLUYtRiRGLUYtKUZARixGNkYtKiYtRj02IyomLCZGQUYtIiImRi1GLUYkRi1GLSlGSUYsRjZGNi9GQzsiIiFGJkYtRi1GJUYlRiU=

Checking: simplify(f3o(t,7)-sinsum(f3,48,t));

IiIh