The FourierSeries Package : Examples of Usage 

Amir H. Khanshan 

E-mail: khanshan@yahoo.com 

Example 1 

Find the fourier series coefficient of the following function(Fig.1). 

>	restart; libname:=libname, currentdir():

 

>	with(FourierSeries);

 

(1.1)

 

>

 

>	f:=piecewise(t<1,t/2+1/2,t>=1,-t+2);

 

(1.2)

 

>	plot(f,t=-5..5);

 

 

>	plot(rept(f,t=-1..2),t=-4..5,thickness=2,tickmarks=[10,2]);

 

 

Fig.1 

>	fs(f,t=-1..2,trig);

 

(1.3)

 

(1.3)

 

>	decompose(a,3);

 

(1.4)

 

>	decompose(b,3);

 

(1.5)

 

>

b(5);

 

(1.6)

 

>

 

exponential form:

 

>	f:=piecewise(t<1,t/2+1/2,t>=1,-t+2):

 

>	plot(rept(f,t=-1..2),t=-4..5,thickness=2,tickmarks=[10,2]);

 

 

>	fs(f,t=-1..2,exp);

 

(1.7)

 

(1.7)

 

>	decompose(c,3);

 

(1.8)

 

>

evalc(%);

 

(1.9)

 

Example2: Demostration of Gibb's Phenomenon 

>	u:=Heaviside:

 

>	rect:=t->u(t+1/2)-u(t-1/2):

 

>	plot(rect(t),t=-3..3,thickness=2);

 

 

>	plot(rept(rect(t),t=-1..1),t=-3..3,thickness=2);

 

 

>	fs(rect(t),t=-1..1,trig,'g'):

 

(2.1)

 

(2.1)

 

>	GP:=N->plot([g(N),rept(rect(t),t=-1..1)],t=-3..3,color=[blue,green],numpoints=10*N):

 

>

GP(3);

 

 

>

GP(5);

 

 

>

GP(10);

 

 

>

 

>

GP(70);

 

 

Example3 

>	plot(rept(exp(t),t=-1..1),t=-5..5,0..3,color=blue,discont=true);

 

 

>	fs(exp(t),t=-1..1,exp);

 

(3.1)

 

(3.1)

 

>	convert(c(n),trig):

 

>	combine(%);

 

(3.2)

 

>

c(0);

 

(3.3)

 

Example4 

>	fs(cos(t)^4,t=0..Pi,trig);

 

(4.1)

 

(4.1)

 

>	cos(t)^4=combine(cos(t)^4);

 

(4.2)

 

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