a r s a b h k _ 5 1 0 2 1_ e m s _ 1 0 1 2 fi l_ u k c a
a r s a b h k _ 5 1 0 2 1_ e m s _ 1 0 1 2 fi l_ u k c a
a r s a b h k _ 5 1 0 2 1_ e m s _ 1 0 1 2 fi l_ u k c a
f (x)
f (x + p) = f (x) f (x) = sin x
x
r a s sin(x + 2π) = sin x a b h T k _ 5 1 0 2 1_ e m s _ 1 0 ωt + sin ωt) = Aeiωt z = x + iy = A(cos 1 2 fi l_ u k c a
p 2π sin
f (x) f (x) x = b f (x)
x
x = a x
2πx T
f (x)
[a, b]
b
ˆ f (x)[a,b] =
f (x)dx
a
b−a
f (x) = sin(x)
sin(x)
[0, π]
[0, π]
1 sin(x)[0,π] = (π)
π
1 sin x dx = (− cos x) π
ˆ 0
π
0
=
1+1 2 = π π
f (x) = sin2 x cos2 x
a r s a y = sin (x) y = cos (x) b h k _ 5 1 0 2 1_ e m x x s _ 1 0 sin 2 ( 1 cos (x) x) fi l_ u [−π, k π] c a ˆ π 2
2
2
sin2 x
2
cos2 x
[−π,π ]
[−π,π ]
=
1 2π
sin2 x dx
π π
−
=
ˆ 1
2π
π
−
cos2 x dx
g(x) =
sin2 x
[−π,π ]
+ cos2 x
π
1 = π,π ] 2π
ˆ
[−
(sin2 x + cos2 x) dx
π π
−
1 = 2π
ˆ
dx = 1
π
−
sin2 x
cos2 x sin2 x [
sin2 x
sin2 nx
= cos2 x π,π ]
[−
= π,π ]
[−
sin2 nx [−π,π ]
= cos2 nx
[−π, π] 2
π,π ]
−
= cos x
1 2
cos2 nx
[−π,π ]
=
1 2
a r s a b h k _ 5 1 1 f (x) = a0 + a1 cos x + a2 cos 0 2x + a3 cos 3x + . . . 2 2 _ + b1 sin x + b2 1 sin 2x + b3 sin 3x + . . . e m s sin nx cos nx 2π _ 1 f (x) 2π 0 1 2 fi π _ ˆ l 1 u nx[ π,π ] = sin mx cos sin mx cos nxdx = 0 k 2π a c π −
−
1 sin mx sin nx[ π,π ] = 2π −
π
ˆ
sin mx sin nxdx
π
−
=
0, 1 2, 0,
m = n m = n =0 m = n = 0
[−π,π ]
1 cos mx cos nx[ π,π ] = 2π −
π
ˆ
cos mx cos nxdx
π
−
=
f (x)
0, 1 2, 1,
m = n m = n =0 m = n = 0 1 2π
[−π, π] 1 2π
π
ˆ π
a0 1 f (x) dx = 2 2π
−
π
ˆ π
1 dx + a1 2π
−
π
ˆ π π
−
+ b1
ˆ 1
2π
π
1 cos x dx + a2 2π
ˆ
cos2x dx + . . .
π
−
sin x dx + . . .
π
−
π
π
f (x) a rdx s a π π b h k a1 f (x) _ 5 1 1 cos x [−π, π] 0 2π 2 _ π π 1 ˆ ˆ ˆ π 2 1 a0 1 m 1 e cos x dx + a1 2π cos x dx f (x)cos x dx = s 2π 2 2π _ π π π 1 0 π 1 1 ˆ 2 + a2 cos2x cos x dx + . . . fi 2π _ l π u π k ˆ 1 c a + b1 sin x cos x dx + . . . 1 2π
ˆ
a0 f (x) dx = 2
=⇒
1 a0 = π
−
ˆ
−
−
−
−
−
2π
π
−
1 2π
π
ˆ π
−
1 f (x)cos x dx = a 1 2π
π
ˆ π
−
cos2 x dx =
a1 2
1 a1 = π
π
ˆ
f (x)cos x dx
π
−
an
1 an = π
π
ˆ
f (x)cos nx dx
π
−
f (x) [−π, π]
1 sin x 2π
bn
1 bn = π
π
ˆ
f (x)sin nx dx
a r s a b h k _ 5 1 0 2 1_ e m 2π s _ 1 0 1 2= 0, −π < x < 0, f (x) fi _ 1, 0 < x < π, l u k c af (x) π
−
an
bn
π
1 an = π
ˆ
f (x)cos nx dx
π
−
=
0
ˆ
1 π
π
ˆ
0 · cos nxdx +
1 · cos nxdx
0
π
−
π
1 = π
1 bn = π
ˆ
cos nxdx =
0
0, 1,
n = 0, n = 0
π
ˆ
f (x)sin nx dx
π
−
=
1 π
0
ˆ
π
0 · sin nxdx +
π
ˆ
1 · sin nxdx
0
−
π
, r a n s 0 a b h k _ 5 1 2 sin x sin3x 1 sin5x 0+ 5 + . . . + f (x) = + 2 2 π 1 3 1_ f (x) n = 3, 9 e m s _ 1 0 y 1 2 fi l_ u k c a 1 = π
ˆ
sin nxdx =
0, 2 , nπ
n
x n = 3 n = 9 n = 15
15
f (x)
n = 3, 9
15
[−π, π ]
n
f (x) • • • •
|f (x)|
x
a rf (x) s a f (x) b h k _ 5 1 0 2 1_ e m s _ 1 0 1 2 fi l_ u k c a einx − e sin nx = 2i
inx
−
,
einx + e cos nx = 2
inx
−
f (x) = c0 + c1 eix + c
+ c2 e2ix + c
ix
1e
−
−
2e
2ix
+ . . .
−
−
n=+∞
=
cn einx
n=−∞
f (x) =
1 a0 + a1 cos x + a2 cos 2x + a3 cos 3x + . . . 2 + b1 sin x + b2 sin 2x + b3 sin 3x + . . .
a0 = + a1 2
eix + e ix e2ix + e + a2 2 2 eix − e ix e2ix − e + b2 2i 2i
+ b1
−
−
f (x) = c 0 + c1 eix + c + c3 e3ix + c
−ix 1e −3ix
−
3e
−
2ix
−
+ . . .
2ix
−
+ c2 e2ix + c
2e
2ix
−
−
+ . . .
n=+∞
a r s a b h cn k _ an bn 5 1 0 ˆ π ikx eikx π 2 ikπ e − e ikπ _ e dx = = =0 ik π 1 ik π e m s _ 1 1 f (x) 0 2π 1 x = −π x = 2 π fi _ l π ˆ ˆ π u 1 k 1 af (x)dx = c0 =⇒ c0 = 2π f (x)dx 2π c π π =
cn einx
n=−∞
−
−
−
−
−
f (x) x = −π
e
inx
−
x = π
+ . . .
1 2π
π
ˆ
f (x)e
inx
−
π
π
1 dx = c 0 2π
−
ˆ
inx
e
−
π
1 dx + c1 2π
−
e
inx ix
e dx
−
π
−
1 + c 1 2π −
π
ˆ
π
ˆ
e
inx
−
e
ix
−
dx + . . .
π
−
cn π
1 cn = 2π
ˆ
inx
f (x)e
−
dx
π
−
2π
0, 1, 0,
[−π, π]
−π < x < 0 0 < x < π2 π 2 < x < π
a r s a b h k _ 5 1 0 2 f (x) _ π ˆ + 1 1 m cn einx cn = f (x)e inx dx e 2π s n= 1_ π f (x) 0 1 2 π/ 2 0 fi ˆ ˆ ˆ π _ 1 l inx inx cn = f (x)e u dx + f (x)e dx + f (x)e k 2π π 0 π/ 2 c a f (x) =
f (x) =
∞
−
−∞
−
−
−
−
π/ 2
1 = 2π
ˆ
e
inx
−
inx
−
dx
dx
0
c0 π/ 2
1 c0 = 2π
ˆ 0
1 dx = 2π
π 1 − 0 = 2 4
cn
n =0
π/ 2
1 cn = 2π = −
ˆ 0
π/ 2
1 e inx dx = − 2inπ −
1 e i2nπ
cn =
inπ/2
−
− 1 =
ˆ
e
inx
−
d(−inx)
0
1 − cos(nπ/2) + i sin(nπ/2) i2nπ
1−i , 2nπ
n = 1, 5, 9, . . .
−2i , 2nπ
n = 2, 6, 10, . . .
−1 − i , 2nπ
n = 3, 7, 11, . . .
0,
n = 4, 8, 12, . . .
a r s a b + h k inx _ f (x) = cn e 5 n= 1 0 ix i2 x 1 1 (1 − i)e 2ie 2 (1 + i)ei3x = + − 1_ − + . . . 4 2π 1 2 3 m2ie i2x (1 + i)e i3x ix e 1 (1 − i)e s + − − + . . . 2π −1 1_ −2 −3 0 1 2 fi l_ 2π u k [−π, a π] 2π c [0, 2π] ∞
−∞
bn
cn
−
−
−
an
2π
1 an = π
ˆ
1 bn = π
ˆ
0 2π
1 cn = 2π
2π
f (x)cos nx dx
f (x)sin nx dx
0 2π
ˆ
f (x)e
inx
−
dx
0
2l
[−l, l]
sin
nπx l
sin
2l
[0, 2l]
nπ nπx nπx (x + 2l) = sin + 2nπ = sin l l l
nπx l f (x)
einπx/l
a r s a 2l b h k 5_ 3πx 1 πx 2πx 1 f (x) = a0 + a1 cos + a2 cos 0+ a3 cos + . . . 2 l l l 2 _ πx 2πx 3πx + b1 sin + b2 sin 1 + b3 sin + . . . m l l l e s _ f (x) 1 0 1 iπx/l iπx/l f (x) = c0 + c1 e + c 2 + c2 e2iπx/l + c 2 e 2iπx/l + . . . 1e fi n=+ l_ inπx/l u = cn k e a = n c cos
2l
−
−
−
−
∞
−∞
−
l
1 2l
ˆ
sin
mπx nπx cos dx l l
=0
l
−
l
ˆ 1
2l
mπx nπx sin sin dx = l l
l
−
1 2l
l
ˆ
mπx nπx cos cos dx = l l
l
−
0, 1 2, 0,
m = n m = n =0 m = n = 0
0, 1 2, 1,
m = n m = n =0 m = n = 0 [−l, l]
l
1 an = l
ˆ
f (x)cos
nπx dx l
f (x)sin
nπx dx l
l
−
1 bn = l
l
ˆ
a r s ˆ l inπx/l h b a 1 cn = f (x)e dx k 2l _ l 5 1 0 2 [0, 2l] _ 1 ˆ 2l e m nπx 1 s an = f (x)cos dx l _ l 1 0 0 1 2 1 ˆ 2l fi _ bn = l f (x)sin nπx dx l l u 0 k c a ˆ 2l 1 l
−
−
−
cn =
f (x)e
inπx/l
−
2l
0
dx
f (x) =
0, 1,
0 < x < l, l < x < 2l,
[0, 2l] cn 2l
1 cn = 2l
ˆ
1 = 2l
ˆ
1 = 2l
f (x)e
inπx/l
−
dx
0 l
0·e
inπx/l
−
0
1 dx + 2l
1·e
inπx/l
−
dx
l
2l
ˆ
e
inπx/l
−
1 e inπx/l dx = 2l −inπ/l
l
2l
ˆ
0,
n
= 0
−
2l l
=
(e
2inπ
−
−e −2inπ
inπ
−
)
a r s a b h k ˆ 2l 1 51_ 1 c0 = f (x)dx = 0 2 2l 2 l 1_ e m s _ 1 3iπx/l 1 3iπx/l 1 1 iπx/l 1 iπx/l f (x) = − e −e + e − e + . . . 0 2 iπ 3 3 1 2 1 3πx fi 1 2 πx = − sin_ + sin + . . . l 2 π u l 3 l k c a =
−
1 , inπ
n
−
f (x) f (x) = x2 f (x) = cos x f (x)
−
f (−x) = f (x)
f (−x) = −f (x) f (x) = x f (x) = sin x f (x) = x 3
l
l
ˆ
f (x) dx =
l
ˆ 2
f (x) dx,
f (x)
0
0,
−
f (x)
f (x)
g(x) h(x) = f (x)g(x) h(−x) = f (−x)g(−x) = −f (x)g(x) = −h(x)
a r s a b nπx h f (x) f (x)cos k l _ 5 1 0 2 l 1_ ˆ 1 m nπx dx = 0 an = f (x)cos e s l l _ l 1 1 0 nπx 2 f (x)sin l fi l_ u k l ˆ ˆ l c a 1 2 nπx nπx −
bn =
f (x)sin
l
l
dx =
l
l
−
f (x)sin
0
f (x)
1 an = l
l
ˆ l
−
l
f (x)cos
nπx 2 f (x)cos dx = l l
l
ˆ 0
f (x)cos
dx
nπx l
nπx dx l
an
bn
f (x)sin
nπx l
1 bn = l
l
ˆ
f (x)sin
nπx dx = 0 l
l
−
an = 0
f (x)
f (x)
l
2 bn = l
ˆ
2 an = l
ˆ
f (x)sin
0
nπx dx l
l
f (x)cos
nπx dx l
a r s a b h k _ 5 1 0 2 1_ e m s _ 1 1 0 an cos nx + bn sin nx f (x) = a0 + 2 1 1 1 2 fi l_ [−π, π] u k [f (x)]2 a c 0
bn = 0
[−π, π]
∞
1 = π,π ] 2π
[f (x)]
1 2 2 an
2
∞
[−
2 1 2 a0 [−π,π ] 2 1 2 (bn sin nx) = 2 bn [−π,π ]
π
ˆ
[f (x)]2 dx
π
−
=
2 1 2 a0
(an cos nx)
2
[−π,π ]
=
2
[f (x)]
ˆ
[−
[f (x)]2 dx
π
−
=
nπ ωn = l
π
1 = π,π ] 2π
2
∞
1 a0 2
[−l, l]
a r s a b h a0 f (x) = + an cos ωn x + k _ bn sin ωnx 2 5 n=1 1n=1 0 2 1_ ˆ l e m 1 s a0 = _ f (τ ) dτ l 1 0 l 1 2 ˆ l fi _ an = 1 f (τ )cos ωnτ dτ l l u k l c a ˆ l ∞
∞
−
−
bn =
1 l
f (τ )sin ωn τ dτ
l
−
∞
1 1 + a2n + b2 2 n=1 2 n=1 n
l
1 f (x) = 2l
ˆ l
−
∞
1 f (τ ) dτ + cos ωn x l n=1
f (τ )cos ωn τ dτ
l
−
∞
1 + sin ωn x l n=1
l
ˆ
f (τ )sin ωn τ dτ
l
−
(n + 1)π nπ π − = l l l
∆ω = ω n+1 − ωn =
1 f (x) = 2l
l
ˆ
1 ∆ω = l π
=⇒
l
ˆ
f (τ ) dτ
l
−
+
∆ω π
∞
cos ωn x
n=1
l
l
ˆ
ˆ
f (τ )cos ωn τ dτ + sin ωn x
l
f (τ )sin ωn τ dτ
l
−
−
a r s a ∞ ∆ω → dω b h+ k + ˆ ˆ _ ˆ 1 5 f (x) = cos ωx f (τ )cos ωτ dτ + 1 sin ωx f (τ )sin ωτ dτ π 0 2 0 1_ e m s _ ˆ + 1 1 0 A(ω) 1 = f (τ )cos ωτ dτ π 2 fi _ l + ˆ u 1 k c a B(ω) = π f (τ )sin ωτ dτ ∞
∞
∞
−∞
−∞
∞
−∞
∞
−∞
∞
f (x) =
ˆ 0
∞
A(ω)cos ωx dω +
ˆ
B(ω)sin ωx dω
0
l →
dω
f (x) f (x) +∞
ˆ
f (−x) = f (x)
+∞
f (x) dx = 2
ˆ
f (x) dx
0
−∞
f (τ )sin ωτ f (x)
B(ω) = 0
∞
f (x) =
ˆ
A(ω)cos ωx dω
0 ∞
2 A(ω) = π
ˆ
f (x)cos ωx dx
0
f (x)
f (−x) = −f (x)
+∞
ˆ
f (x) dx = 0
−∞
A(ω) = 0
f (x)cos ωx
a r s a ˆ b h f (x) = B(ω)sin ωx k dω _ 0 5 1 ˆ 0 2 2 B(ω) = f (x)sin _ ωx dx π 1 0 m e s _ 1 0 1 2 ˆ + fi _ lf (x) = C (ω)eiωx dω u k c a ˆ + ∞
∞
∞
−∞
∞
1 C (ω) = 2π
f (x)e
iωx
−
dx
−∞
f (x) =
2
π
∞
´ f (x)cos ωx dx 0
2
π
∞
´ A(ω)cos ωx dω 0
A(ω ) =
f (x) =
1, 0,
−1 < x < 1 |x| > 1
f (x) 1
∞
2 A(ω) = π
ˆ 0
2 f (x)cos ωx dx = π
ˆ
cos ωx dx =
0
2 [sin ω − 0] ωπ
2 sin ω = π ω ∞
f (x) =
ˆ
A(ω)cos ωx dω
a r s a = cos ωx b dω π ω h k 0 _ 5 1 0 2 1_ e m s _ sin2x, 0 ≤ x ≤ π 1 f (x) = 0 1 0, 2 fi _ l f (x) u k a c B(ω) 0
∞
ˆ sin ω 2
A(ω)
+∞
1 A(ω) = π
1 f (x)cos ωx dx = π
ˆ
−∞
π
=
π
0
1 = 4πi =
=
i2x
ˆ e 1
1 4πi
−
sin2x cos ωx dx
0
eiωx + e 2
iωx
−
dx
π
ˆ
i(2+ω )x
e
+ e
i(2−ω )x
i(2−ω )x
−e
−
−e
i(2+ω )x
−
0 π
ˆ 0 π
ˆ 1
2π
i2x
−e 2i
π
ˆ
0
dx
(2i(sin(2 + ω)x) + 2i(sin(2 − ω)x)) dx
1 sin(2 + ω)x dx + 2π
π
ˆ
sin(2 − ω)x dx
0
cos(2 + ω)x cos(2 − ω)x π − 2π(2 + ω) 0 2π(2 − ω) 0 cos(2 + ω)π 1 cos(2 − ω)π 1 − = − + + 2π(2 + ω) 2π(2 + ω) 2π(2 − ω) 2π(2 − ω) 2 = (1 − cos ωπ) π(4 − ω 2 )
= −
π
a r s a b h k ˆ + ˆ π 1 5_ 1 1 B(ω) = f (x)sin ωx dx = sin2x sin ωx dx 0 π π 2 10 _ π ˆ m 1 ei2x − e i2x e eiωx − e iωx = dx s _ π 2i 2i 1 0 0 1 2sin ωπ 2 = 2 π(4 − ω ) fi l_ u k c a ∞
−∞
−
−
f (x) f (x) =
1, 0,
− π < x < 0 0 < x < π
− π < x < 0 π −1, 0 < x < 2 π 1, < x < π 2 π + x, − π < x < 0 π − x, 0 < x < π
2π f (x) =
0,
0, 1,
π − π
a r− s a b h k _ 5 f (x) = 1 0 2 1_ e m s _ 1 0 1 + 2x, 1 2 f (x) = x, −1 < x < 1 f (x) = fi 1 − 2x, _ l 0, k − u1 < x < 0, x/2, f (x) = f (x) = a 1, c 0 < x < 3. 1, f (x) =
f (x) =
0, x,
π < x < 0 0 < x < π
− 1 < x < 0, 0 < x < 1. 0 < x < 2, 2 < x < 3.
f (x) f (x) =
1, 0,
|x| < a, |x| > a.
∞
1 − x2 , 0,
|x| < 1, |x| > 1.
∞
ˆ sin αa cos αx α
−∞
f (x) =
dx
ˆ x cos x − sin x 0
x3
cos(x/2)dx
a r s a b h k _ 5 1 0 2 1_ e m s _ 1 0 1 2 fi l_ u k c a
