(仅供参考)常见函数的傅里叶级数

Fig. 24-12
Miscellaneous Fourier Series
24.19.
f (x) = sin μx, − π < x < π, μ ≠ integer
2 sin μπ π
⎛ sin x ⎝⎜12 − μ2
−
2 sin 2x 22 − μ2
+
3sin 3x 32 − μ2
−
⎞ ⎠⎟
24.20.
α π
−
2 π
⎛ sinα cos ⎝⎜ 1
x
−
sin 2α cos 2x 2
+
sin
3α cos 3
3x
−
⎞ ⎠⎟
24.18.
f
(x)
=
⎧⎨⎩−
x(π x(π
− −
x x
) )
0< x <π −π < x < 0
8 π
⎛ sin x ⎝⎜ 13
+
sin 3x 33
+
sin 5x 53
+
⎞ ⎠⎟
Fig. 24-11
∑ 24.1.
a0 2
+
∞ n=1
⎛ ⎝⎜
an
cos
nπx L
+ bn
sin
n π x⎞ L ⎠⎟
where
24.2.
∫ ⎧⎪an
⎨
∫ ⎩⎪bn
= =
1
L 1
L
c+2 L
nπx
c f (x) cos L dx
c+2 L
nπx
c f (x)sin L dx
If f (x) and f ′(x) are piecewise continuous and f (x) is defined by periodic extension of period 2L, i.e.,
π2 3
−4⎛ ⎝⎜来自cos 12x
−
cos 2x 22
+
cos 3x 32
−
⎞ ⎠⎟
24.15. f (x) = x(π − x), 0 < x < π
π2 6
−
⎛ ⎝⎜
cos 2 12
x
+
cos 4x 22
+
cos 6x 32
+
⎞ ⎠⎟
24.16. f (x) = x(π − x)(π + x), − π < x < π
Assuming that the series 24.1 converges to f (x), we have
24.3.
∞
∑ f (x) = cneinπx/L n=−∞
where
24.4.
cn
=
1 2L
∫ c+2L c
f (x)e−inπx/L dx
=
⎧⎨⎪1212 ⎩⎪12
(an − ibn ) (a−n + ib−n ) a0
cos 5x 52
+
⎞ ⎠⎟
24.9. f (x) = x, − π < x < π
2
⎛ sin ⎝⎜ 1
x
−
sin 2x 2
+
sin 3x 3
−
⎞ ⎠⎟
24.10. f (x) = x, 0 < x < 2π
π
−
2
⎛ sin ⎝⎜ 1
x
+
sin 2x 2
+
sin 3x 3
+
⎞ ⎠⎟
24.11. f (x) = | sin x |, − π < x < π
(ancn
n=1
+ bndn )
where an, bn and cn, dn are the Fourier coefficients corresponding to f (x) and g(x), respectively.
Special Fourier Series and Their Graphs
24.7.
f (x) = ⎧⎨⎩−11
0< x <π −π < x < 0
4 π
⎛ sin ⎝⎜ 1
x
+
sin 3x 3
+
sin 5x 5
+
⎞ ⎠⎟
24.8.
f (x) = | x | = ⎧⎨⎩−xx
0< x <π −π < x < 0
π 2
−
4 π
⎛ cos x ⎝⎜ 12
+
cos 3x 32
+
Definition of a Fourier Series
The Fourier series corresponding to a function f (x) defined in the interval c Ϲ x Ϲ c + 2L where c and L > 0 are constants, is defined as
f (x) = cos μx, − π < x < π, μ ≠ integer
2μ sin μπ π
⎛1 ⎝⎜ 2μ2
+
cos x 12 − μ2
−
cos 2x 22 − μ2
+
cos 3x 32 − μ2
−
⎞ ⎠⎟
24.21. 24.22.
f (x) = tan−1[(a sin x) / (1 − a cos x)], − π < x < π , | a | < 1
a sin x + a2 sin 2x + a3 sin 3x +
12
⎛ ⎝⎜
sin 13
x
−
sin 2x 23
+
sin 3x 33
−
⎞ ⎠⎟
Fig. 24-6 Fig. 24-7 Fig. 24-8 Fig. 24-9 Fig. 24-10
24.17.
f (x) = ⎧⎨⎪10
0< x <π −α π −α < x <π +α
⎩⎪0 π + α < x < 2π
f (x
ϩ
2L)
ϭ
f (x),
then
the
series
converges
to
f (x)
if
x
is
a
point
of
continuity
and
to
1 2
{
f
(x
+
0)
+
f (x
−
0)}
if
x is a point of discontinuity.
Complex Form of Fourier Series
n>0 n<0 n=0
Parseval’s Identity
24.5.
∫ ∑ 1
L
c+2 L
{ f (x)}2 dx
c
=
a02 2
+
∞
(an2
n=1
+ bn2 )
Generalized Parseval Identity
∫ ∑ 24.6.
1 L
c+2 L c
f (x)g(x) dx
=
a0c0 2
+
∞
x
−
2 π
⎛⎝⎜
cos 2x 1i 3
+
cos 4x 3i5
+
cos 6x 5i7
+
⎞⎠⎟
24.13.
f
(x)
=
⎧⎨⎩−
cos cos
x x
0< x <π −π < x < 0
8 π
⎛ sin 2x ⎝⎜ 1 i 3
+
2 sin 4x 3i5
+
3sin 6x 5i7
+
⎞⎠⎟
24.14. f (x) = x2, − π < x < π
2 π
−
4 π
⎛ cos 2x ⎝⎜ 1 i 3
+
cos 4x 3i5
+
cos 6x 5i7
+
⎞⎠⎟
Fig. 24-1 Fig. 24-2 Fig. 24-3 Fig. 24-4 Fig. 24-5
24.12.
f (x) = ⎧⎨⎩sin0 x
0< x <π π < x < 2π
1 π
+
1 2
sin