時間関数(周期T):1周期分
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フーリエ級数 (周期 T=2π)
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矩形波
(奇関数)
f(x) = 0 (-π≦ x < 0 )
= 1 ( 0 ≦ x ≦π )
T = 2π(-π〜π)
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積分区間:[-π→π]
a0 = 1/π∫f(x)dx = 1
an = 1/π ∫f(x)cos(nx)dx = 0 (n≧1)
bn = 1/π ∫f(x)sin(nx)dx = 2/(nπ) (n:odd),
= 0 (n:even)
f(x) = 1/2+(2/π){sin(x)+(1/3)sin(3x)+(1/5)sin(5x)+…}
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矩形波(奇関数)
f(x) = -1 (-π≦ x < 0 )
= 1 ( 0 ≦ x ≦π )
T = 2π(-π〜π)
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T=2π,ωo=2π/T=1
an=0(n=0,1,2,…)
bn=(2/T)∫[-T/2,T/2] f(x)sin(2nπt/T)dt
=(2/π)∫[0,π] sin(nt) dt
=(2/π)[-cos(nx)/n][0,π] =(2/π){1-cos(nπ)}
=(4/(nπ))(n=2m-1), =0(n=2m)
f(t)=(4/π)Σ[m=1,n] sin((2m-1)x)/(2m-1) (n→∞)
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矩形波(偶関数)
f(x) = 0 (π/2≦ | x | ≦ π)
= 1 ( | x | ≦π/2 )
T = 2π(-π〜π)
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積分区間:[-π→π]
a0 = 1/π∫f(x)dx = 1
an = 1/π ∫f(x)cos(nx)dx = -{(-1)^n}*2/(nπ) (n:odd),
= 0 (n:even)
bn = 1/π ∫f(x)sin(nx)dx = 0
f(x) = 1/2+(2/π) { sin(x)-(1/3)sin(3x)+(1/5)sin(5x)-…} |
鋸波
f(x) = x (-π≦x ≦π)
T = 2π(-π〜π)
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積分区間:[-π→π]
an = (1/π) ∫x cos(nx)dx = 0
bn = (1/π) ∫x sin(nx)dx
= (2/π) ∫x sin(nx)dx (積分区間:[ 0 →π] )
= (-1)^(n+1) 2/n
f(x) = 2{ sin(x)-(1/2)sin(2x)+(1/3)sin(3x)-・・・)
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f(t)=t (t=-π〜π)
フーリエ級数展開係数はf(t)が奇関数であるから
an=0 (n=0,1,2,…)
bn=(4/T)∫(0〜T/2) t sin(nωot) (n=1,2,3,…)
=(2/π)∫(0〜π) t sin(nt)dt
部分積分して
bn=(2/π){[t(-1/n)cos(nt)](t:0〜π)+(1/n)∫(0〜π) cos(nt)dt}
=(2/π){-(π/n)cos(nπ)+(1/n)[(1/n)sin(nt)(t:0〜π)}
=(2/π){-(π/n)*(-1)^n+((1/n)^2)sin(nπ)}
=(2/n)*(-1)^(n+1) (n=1,2,3,…)
フーリエ級数展開は
f(t)=Σ(n=1,∞) (2/n)*((-1)^(n+1))sin(nt)
となる。
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放物線y=(π^2)- (x^2) (-π≦x ≦π)
T = 2π(-π〜π)
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y=(2/3)π^2
+4cos(x)-cos(2x)+(4/9)cos(3x)-(1/4)cos(4x)+(4/25)cos(5x)-…
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放物線y = x^2 (-π≦x ≦π)
T = 2π(-π〜π)
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y=(1/3)(π^2)-4cos(x)+cos(2x)-(4/9)cos(3x)+(1/4)cos(4x)-(4/25)cos(5x)+…
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y=cos (x/2) (-π≦x≦π)
T=2π(-π〜π)
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[Maple10]
>
> an=
>
= (-1)^(n+1)
*(4/π)(1/((4n^2)-1))
>
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フーリエ級数
展開 積分区間[0〜1]
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周期 T=1
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y=f(x)=x2
(0≦x≦1), f(x)=f(x+1)
To=1,ωo=2π
f(x)=a0/2 +Σ[n=1,∞] {an cos(nωo x)+bn
sin(nωo x)}
a0=1/To)∫[0,To] f(x)dx
an=(1/To)∫[0,To] f(x)cos(nωo x)dx
bn=(1/To)∫[0,To] f(x)sin(nωo x)dx
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[Maple10]
a0=∫[0,1] x2dx = x3/3| [0,1]=1/3
an=∫[0,1] x2cos (2nπx) dx =1/(2π2
n2 )
bn=∫[0,1] x2sin (2nπx) dx=-1/(2nπ)
f(x)=(1/6)
+Σ[n=1,∞] [{1/(2π2n2)}
cos(2nπx)-{1/(2nπ)}sin(2nπx)]
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フーリエ級数
展開 積分区間 [-1〜1] |
周期 T=2 |
y=f(x)=0 (-1≦x<0), x (0≦x≦1
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[Maple10]
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