# funkcia
f1(x)=-5/2*x
f2(x)=-1/2
f3(x)=25/4*x-3
f4(x)=3/4
f5(x)=-15/4*x+15/4
plot(f1,(x,0,1/5))+plot(f2,(x,1/5,2/5))+plot(f3,(x,2/5,3/5))+plot(f4,(x,3/5,4/5))+plot(f5,(x,4/5,1))
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# Koeficienty Kosinusovy rad
a(n)=2*(integral(f1(x)*cos(n*pi*x),(x,0,1/5))+integral(f2(x)*cos(n*pi*x),(x,1/5,2/5))+integral(f3(x)*cos(n*pi*x),(x,2/5,3/5))+integral(f4(x)*cos(n*pi*x),(x,3/5,4/5))+integral(f5(x)*cos(n*pi*x),(x,4/5,1)))
a0=2*(integral(f1(x),(x,0,1/5))+integral(f2(x),(x,1/5,2/5))+integral(f3(x),(x,2/5,3/5))+integral(f4(x),(x,3/5,4/5))+integral(f5(x),(x,4/5,1)))
pretty_print(a0)
pretty_print(a(n))
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(\frac{1}{5} \, \pi n\right)}{\pi n} - \frac{\sin\left(\frac{2}{5} \, \pi n\right)}{\pi n} - \frac{3 \, \sin\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi n} + \frac{3 \, \sin\left(\frac{4}{5} \, \pi n\right)}{2 \, \pi n} - \frac{3 \, {\left(\pi n \sin\left(\frac{4}{5} \, \pi n\right) - 5 \, \cos\left(\frac{4}{5} \, \pi n\right)\right)}}{2 \, \pi^{2} n^{2}} + \frac{3 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{2 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} - \frac{\pi n \sin\left(\frac{1}{5} \, \pi n\right) + 5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{\pi^{2} n^{2}} - \frac{15 \, \cos\left(\pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{5}{\pi^{2} n^{2}} \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(\frac{1}{5} \, \pi n\right)}{\pi n} - \frac{\sin\left(\frac{2}{5} \, \pi n\right)}{\pi n} - \frac{3 \, \sin\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi n} + \frac{3 \, \sin\left(\frac{4}{5} \, \pi n\right)}{2 \, \pi n} - \frac{3 \, {\left(\pi n \sin\left(\frac{4}{5} \, \pi n\right) - 5 \, \cos\left(\frac{4}{5} \, \pi n\right)\right)}}{2 \, \pi^{2} n^{2}} + \frac{3 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{2 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} - \frac{\pi n \sin\left(\frac{1}{5} \, \pi n\right) + 5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{\pi^{2} n^{2}} - \frac{15 \, \cos\left(\pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{5}{\pi^{2} n^{2}} |
# Koeficienty Sinusovy rad
b(n)=2*(integral(f1(x)*sin(n*pi*x),(x,0,1/5))+integral(f2(x)*sin(n*pi*x),(x,1/5,2/5))+integral(f3(x)*sin(n*pi*x),(x,2/5,3/5))+integral(f4(x)*sin(n*pi*x),(x,3/5,4/5))+integral(f5(x)*sin(n*pi*x),(x,4/5,1)))
pretty_print(a(n))
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(\frac{1}{5} \, \pi n\right)}{\pi n} - \frac{\sin\left(\frac{2}{5} \, \pi n\right)}{\pi n} - \frac{3 \, \sin\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi n} + \frac{3 \, \sin\left(\frac{4}{5} \, \pi n\right)}{2 \, \pi n} - \frac{3 \, {\left(\pi n \sin\left(\frac{4}{5} \, \pi n\right) - 5 \, \cos\left(\frac{4}{5} \, \pi n\right)\right)}}{2 \, \pi^{2} n^{2}} + \frac{3 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{2 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} - \frac{\pi n \sin\left(\frac{1}{5} \, \pi n\right) + 5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{\pi^{2} n^{2}} - \frac{15 \, \cos\left(\pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{5}{\pi^{2} n^{2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{\sin\left(\frac{1}{5} \, \pi n\right)}{\pi n} - \frac{\sin\left(\frac{2}{5} \, \pi n\right)}{\pi n} - \frac{3 \, \sin\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi n} + \frac{3 \, \sin\left(\frac{4}{5} \, \pi n\right)}{2 \, \pi n} - \frac{3 \, {\left(\pi n \sin\left(\frac{4}{5} \, \pi n\right) - 5 \, \cos\left(\frac{4}{5} \, \pi n\right)\right)}}{2 \, \pi^{2} n^{2}} + \frac{3 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{2 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{2 \, \pi^{2} n^{2}} - \frac{\pi n \sin\left(\frac{1}{5} \, \pi n\right) + 5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{\pi^{2} n^{2}} - \frac{15 \, \cos\left(\pi n\right)}{2 \, \pi^{2} n^{2}} + \frac{5}{\pi^{2} n^{2}}
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# kosinusovy rad useknuty v n=5
festcos(x)=a0/2+sum(a(n)*cos(n*pi*x) for n in range(1,5))
plot(festcos(x),(x,-1,1),color="red")+plot(f1,(x,0,1/5))+plot(f2,(x,1/5,2/5))+plot(f3,(x,2/5,3/5))+plot(f4,(x,3/5,4/5))+plot(f5,(x,4/5,1))
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# sinusovy rad useknuty v n=5
festsin(x)=sum(b(n)*sin(n*pi*x) for n in range(1,5))
plot(festsin(x),(x,-1,1),color="red")+plot(f1,(x,0,1/5))+plot(f2,(x,1/5,2/5))+plot(f3,(x,2/5,3/5))+plot(f4,(x,3/5,4/5))+plot(f5,(x,4/5,1))
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# funkcia periodicky rozsirena na interval [-1,1]
f1(x)=-5/2*x
f2(x)=-1/2
f3(x)=25/4*x-3
f4(x)=3/4
f5(x)=-15/4*x+15/4
f01(x)=-5/2*(x+1)
f02(x)=-1/2
f03(x)=25/4*(x+1)-3
f04(x)=3/4
f05(x)=-15/4*(x+1)+15/4
plot(f01,(x,-1,-4/5))+plot(f02,(x,-4/5,-3/5))+plot(f03,(x,-3/5,-2/5))+plot(f04,(x,-2/5,-1/5))+plot(f05,(x,-1/5,0))+plot(f1,(x,0,1/5))+plot(f2,(x,1/5,2/5))+plot(f3,(x,2/5,3/5))+plot(f4,(x,3/5,4/5))+plot(f5,(x,4/5,1))
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# Fourierove koeficienty
A(n)=integral(f01(x)*cos(n*pi*x),(x,-1,-4/5))+integral(f02(x)*cos(n*pi*x),(x,-4/5,-3/5))+integral(f03(x)*cos(n*pi*x),(x,-3/5,-2/5))+integral(f04(x)*cos(n*pi*x),(x,-2/5,-1/5))+integral(f05(x)*cos(n*pi*x),(x,-1/5,0))+integral(f1(x)*cos(n*pi*x),(x,0,1/5))+integral(f2(x)*cos(n*pi*x),(x,1/5,2/5))+integral(f3(x)*cos(n*pi*x),(x,2/5,3/5))+integral(f4(x)*cos(n*pi*x),(x,3/5,4/5))+integral(f5(x)*cos(n*pi*x),(x,4/5,1))
A0=integral(f01(x),(x,-1,-4/5))+integral(f02(x),(x,-4/5,-3/5))+integral(f03(x),(x,-3/5,-2/5))+integral(f04(x),(x,-2/5,-1/5))+integral(f05(x),(x,-1/5,0))+integral(f1(x),(x,0,1/5))+integral(f2(x),(x,1/5,2/5))+integral(f3(x),(x,2/5,3/5))+integral(f4(x),(x,3/5,4/5))+integral(f5(x),(x,4/5,1))
B(n)=integral(f01(x)*sin(n*pi*x),(x,-1,-4/5))+integral(f02(x)*sin(n*pi*x),(x,-4/5,-3/5))+integral(f03(x)*sin(n*pi*x),(x,-3/5,-2/5))+integral(f04(x)*sin(n*pi*x),(x,-2/5,-1/5))+integral(f05(x)*sin(n*pi*x),(x,-1/5,0))+integral(f1(x)*sin(n*pi*x),(x,0,1/5))+integral(f2(x)*sin(n*pi*x),(x,1/5,2/5))+integral(f3(x)*sin(n*pi*x),(x,2/5,3/5))+integral(f4(x)*sin(n*pi*x),(x,3/5,4/5))+integral(f5(x)*sin(n*pi*x),(x,4/5,1))
pretty_print(A0)
pretty_print(A(n))
pretty_print(B(n))
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\sin\left(\frac{1}{5} \, \pi n\right)}{4 \, \pi n} + \frac{\sin\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi n} - \frac{\sin\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi n} + \frac{\sin\left(\frac{4}{5} \, \pi n\right)}{4 \, \pi n} - \frac{\pi n \sin\left(\frac{4}{5} \, \pi n\right) - 5 \, \cos\left(\frac{4}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{2 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{3 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{2 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{3 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{\pi n \sin\left(\frac{1}{5} \, \pi n\right) + 5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{5 \, \cos\left(\pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{5}{4 \, \pi^{2} n^{2}} \newcommand{\Bold}[1]{\mathbf{#1}}-\frac{5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{4 \, \pi n} + \frac{5 \, \cos\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi n} + \frac{5 \, \cos\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi n} - \frac{5 \, \cos\left(\frac{4}{5} \, \pi n\right)}{4 \, \pi n} + \frac{5 \, {\left(\pi n \cos\left(\frac{4}{5} \, \pi n\right) + 5 \, \sin\left(\frac{4}{5} \, \pi n\right)\right)}}{4 \, \pi^{2} n^{2}} - \frac{2 \, \pi n \cos\left(\frac{3}{5} \, \pi n\right) - 25 \, \sin\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{3 \, \pi n \cos\left(\frac{3}{5} \, \pi n\right) - 25 \, \sin\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{2 \, \pi n \cos\left(\frac{2}{5} \, \pi n\right) + 25 \, \sin\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{3 \, \pi n \cos\left(\frac{2}{5} \, \pi n\right) + 25 \, \sin\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{5 \, {\left(\pi n \cos\left(\frac{1}{5} \, \pi n\right) - 5 \, \sin\left(\frac{1}{5} \, \pi n\right)\right)}}{4 \, \pi^{2} n^{2}} - \frac{25 \, \sin\left(\pi n\right)}{4 \, \pi^{2} n^{2}} \newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{5} \newcommand{\Bold}[1]{\mathbf{#1}}-\frac{\sin\left(\frac{1}{5} \, \pi n\right)}{4 \, \pi n} + \frac{\sin\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi n} - \frac{\sin\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi n} + \frac{\sin\left(\frac{4}{5} \, \pi n\right)}{4 \, \pi n} - \frac{\pi n \sin\left(\frac{4}{5} \, \pi n\right) - 5 \, \cos\left(\frac{4}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{2 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{3 \, \pi n \sin\left(\frac{3}{5} \, \pi n\right) + 25 \, \cos\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{2 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{3 \, \pi n \sin\left(\frac{2}{5} \, \pi n\right) - 25 \, \cos\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{\pi n \sin\left(\frac{1}{5} \, \pi n\right) + 5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{5 \, \cos\left(\pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{5}{4 \, \pi^{2} n^{2}} \newcommand{\Bold}[1]{\mathbf{#1}}-\frac{5 \, \cos\left(\frac{1}{5} \, \pi n\right)}{4 \, \pi n} + \frac{5 \, \cos\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi n} + \frac{5 \, \cos\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi n} - \frac{5 \, \cos\left(\frac{4}{5} \, \pi n\right)}{4 \, \pi n} + \frac{5 \, {\left(\pi n \cos\left(\frac{4}{5} \, \pi n\right) + 5 \, \sin\left(\frac{4}{5} \, \pi n\right)\right)}}{4 \, \pi^{2} n^{2}} - \frac{2 \, \pi n \cos\left(\frac{3}{5} \, \pi n\right) - 25 \, \sin\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{3 \, \pi n \cos\left(\frac{3}{5} \, \pi n\right) - 25 \, \sin\left(\frac{3}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{2 \, \pi n \cos\left(\frac{2}{5} \, \pi n\right) + 25 \, \sin\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} - \frac{3 \, \pi n \cos\left(\frac{2}{5} \, \pi n\right) + 25 \, \sin\left(\frac{2}{5} \, \pi n\right)}{4 \, \pi^{2} n^{2}} + \frac{5 \, {\left(\pi n \cos\left(\frac{1}{5} \, \pi n\right) - 5 \, \sin\left(\frac{1}{5} \, \pi n\right)\right)}}{4 \, \pi^{2} n^{2}} - \frac{25 \, \sin\left(\pi n\right)}{4 \, \pi^{2} n^{2}} |
# Fourierov rad useknuty v n=5
fest(x)=A0/2+sum(A(n)*cos(n*pi*x)+B(n)*sin(n*pi*x) for n in range(1,5))
plot(fest(x),(x,-1,1),color="red")+plot(f01,(x,-1,-4/5))+plot(f02,(x,-4/5,-3/5))+plot(f03,(x,-3/5,-2/5))+plot(f04,(x,-2/5,-1/5))+plot(f05,(x,-1/5,0))+plot(f1,(x,0,1/5))+plot(f2,(x,1/5,2/5))+plot(f3,(x,2/5,3/5))+plot(f4,(x,3/5,4/5))+plot(f5,(x,4/5,1))
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