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π
k
()
cos
dxkxxf
π−
π
a
0
2
π−
n
∞
¦
π
§
¨
³³³³
¨
1
=
π−
©
π
++=
nn
π−
·
¸
.cossincoscoscos
dxkxnxbdxkxnxadxkx
¸
¹
ȼɫɟ ɢɧɬɟɝɪɚɥɵ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ, ɤɪɨɦɟ ɢɧɬɟɝɪɚɥɚ ɩɪɢ ɤɨɷɮɮɢɰɢɟɧɬɟ
ɧɭɥɸ ɜɫɥɟɞɫɬɜɢɟ ɮɨɪɦɭɥ 2), 3) ɢ 4). ɂɧɬɟɝɪɚɥ ɠɟ ɩɪɢ
ɪɚɜɟɧ π.
a
k
ɪɚɜɧɵ
a
k
Ɂɧɚɱɢɬ,
π
()
³³
π−
π
2
coscos
π−
.
π==
adxkxadxkxxf
kk
Ɉɬɫɸɞɚ ɧɚɯɨɞɢɦ
:
a
k
π
1
= dxkxxfa
k
()
cos
³
π
π−
. (8.5)
Ɏɨɪɦɭɥɚ (8.4) ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɮɨɪɦɭɥɵ (8.5) ɩɪɢ
ɑɬɨɛɵ ɧɚɣɬɢ ɤɨɷɮɮɢɰɢɟɧɬ
ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɜ ɩɪɟɞɟɥɚɯ
, ɭɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜɚ (8.3) ɧɚ
b
k
π−
ɞɨ π:
0=
.
kxsin
π
()
sin
dxkxxf
π−
π
a
0
2
π−
¦
n
π
∞
§
¨
¨
=
1
π−
©
π
++=
nn
³³³³
π−
·
¸
.sinsinsincossin
dxkxnxbdxkxnxadxkx
¸
¹
ȼ ɫɢɥɭ ɮɨɪɦɭɥ 1), 3) ɢ 5) ɜɫɟ ɢɧɬɟɝɪɚɥɵ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɪɚɜɧɵ ɧɭɥɸ, ɤɪɨɦɟ
ɢɧɬɟɝɪɚɥɚ ɩɪɢ ɤɨɷɮɮɢɰɢɟɧɬɟ
. ɗɬɨɬ ɢɧɬɟɝɪɚɥ ɪɚɜɟɧ π. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
b
k
π
³
()
π−
π=kbdxkxxf sin
ɢ
π
1
= dxkxxfb
k
()
sin
³
π
π−
. (8.6)
()
ɉɭɫɬɶ ɬɟɩɟɪɶ
xf
– ɩɪɨɢɡɜɨɥɶɧɚɹ ɮɭɧɤɰɢɹ, ɡɚɞɚɧɧɚɹ ɜ ɢɧɬɟɪɜɚɥɟ
()
ππ− ,
ɨɬɧɨɫɢɬɟɥɶɧɨ ɤɨɬɨɪɨɣ ɦɵ ɬɨɥɶɤɨ ɩɪɟɞɩɨɥɚɝɚɟɦ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɢɧɬɟɝɪɚɥ ɨɬ ɧɟɟ
()
ɜ ɢɧɬɟɪɜɚɥɟ
()
. ɉɪɢ ɷɬɨɦ ɮɭɧɤɰɢɹ
ππ− ,
xf
ɦɨɠɟɬ ɢɦɟɬɶ ɬɨɱɤɢ ɪɚɡɪɵɜɚ.
,
121
y
0
Ɋɢɫ. 8.1
Ʉɨɷɮɮɢɰɢɟɧɬɚɦɢ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
()
xf
ɧɚɡɵɜɚɸɬɫɹ ɱɢɫɥɚ
x
ɨɩɪɟɞɟɥɹɟɦɵɟ ɮɨɪɦɭɥɚɦɢ
π
1
=
() ()
π
π−
,cos
π
1
=
nn
π
sin
³³
π−
. (8.7)
dxnxxfbdxnxxfa
Ɋɹɞ
∞
a
0
¦
2
1
n
=
++
sincos
nxbnxa
nn
()
xf
ɧɚɡɵɜɚɟɬɫɹ ɪɹɞɨɦ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
ɋɭɦɦɚ ɪɹɞɚ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
π2
. ɉɨɷɬɨɦɭ, ɟɫɥɢ ɪɹɞ ɫɯɨɞɢɬɫɹ ɜ ɢɧɬɟɪɜɚɥɟ
ɨɫɬɚɥɶɧɵɯ ɡɧɚɱɟɧɢɹɯ
x
ɢ ɫɭɦɦɚ ɟɝɨ ɩɟɪɢɨɞɢɱɟɫɤɢ ɩɨɜɬɨɪɹɟɬ ɬɟ ɡɧɚɱɟɧɢɹ, ɤɨɬɨ-
ɪɵɟ ɨɧɚ ɩɪɢɧɢɦɚɥɚ ɜ ɨɫɧɨɜɧɨɦ ɢɧɬɟɪɜɚɥɟ
Ɏɭɪɶɟ ɛɭɞɟɬ ɩɪɟɞɫɬɚɜɥɹɬɶ ɮɭɧɤɰɢɸ
.
()
xf
ɟɫɬɶ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɫ ɩɟɪɢɨɞɨɦ
()
()
()
xf
, ɡɚɞɚɧɧɭɸ ɜ ɢɧɬɟɪɜɚɥɟ
, ɬɨ ɨɧ ɫɯɨɞɢɬɫɹ ɢ ɩɪɢ ɜɫɟɯ
ππ− ,
. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɭɦɦɚ ɪɹɞɚ
ππ− ,
()
ɨɞɢɱɟɫɤɢ ɩɪɨɞɨɥɠɟɧɧɭɸ ɧɚ ɜɫɸ ɱɢɫɥɨɜɭɸ ɨɫɶ.
ɉɨɷɬɨɦɭ, ɝɨɜɨɪɹ ɜ ɞɚɥɶɧɟɣɲɟɦ ɨ ɪɚɡɥɨɠɟɧɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ, ɡɚɞɚɧ-
ɧɨɣ ɜ ɢɧɬɟɪɜɚɥɟ
()
ɮɭɧɤɰɢɢ. ȿɫɥɢ ɩɪɢ ɷɬɨɦ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ
ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ, ɬ. ɟ.
(ɪɢɫ. 8.1), ɚ ɟɫɥɢ
, ɜɫɟɝɞɚ ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɪɟɱɶ ɢɞɟɬ ɨ ɩɟɪɢɨɞɢɱɟɫɤɨɣ
ππ− ,
()
xf
ɧɚ ɤɨɧɰɚɯ ɨɫɧɨɜɧɨɝɨ ɢɧɬɟɪɜɚɥɚ
() ()
() ()
π≠π− ff
π=π− ff
, ɬɨ ɮɭɧɤɰɢɹ ɩɪɨɞɨɥɠɚɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨ
, ɬɨ ɩɪɢ ɬɚɤɨɦ ɩɪɨɞɨɥɠɟɧɢɢ ɤɨɧɰɵ ɨɫɧɨɜɧɨɝɨ ɢɧ-
ɬɟɪɜɚɥɚ ɛɭɞɭɬ ɹɜɥɹɬɶɫɹ ɬɨɱɤɚɦɢ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ (ɪɢɫ. 8.2).
y
ɢ
a
n
ɢ ɩɟɪɢ-
ππ− ,
,
b
n
0
Ɋɢɫ. 8.2
122
x
ȿɳɟ ɧɟɥɶɡɹ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɨɛɪɚɡɨɜɚɧɧɵɣ ɪɹɞ Ɏɭɪɶɟ ɫɯɨɞɢɬɫɹ ɢ ɱɬɨ ɟɝɨ
()
ɫɭɦɦɚ (ɟɫɥɢ ɨɧ ɫɯɨɞɢɬɫɹ) ɪɚɜɧɚ ɮɭɧɤɰɢɢ
ɍɫɥɨɜɢɟ ɪɚɡɥɨɠɟɧɢɹ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ.
ɋɮɨɪɦɭɥɢɪɭɟɦ ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
ɢɦɟɟɬ ɫɜɨɟɣ ɫɭɦɦɨɣ ɢɦɟɧɧɨ ɮɭɧɤɰɢɸ
ȼɜɟɞɟɦ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨ ɧɟɫɤɨɥɶɤɨ ɨɩɪɟɞɟɥɟɧɢɣ.
()
xf
Ɏɭɧɤɰɢɹ
ɜɚɥɟ ɨɧɚ ɧɟɩɪɟɪɵɜɧɚ ɜɦɟɫɬɟ ɫɨ ɫɜɨɟɣ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ
Ɏɭɧɤɰɢɹ
ɜɚɥ
ɤɨɬɨɪɵɯ
ɭɝɥɨɜɵɯ ɬɨɱɟɤ, ɧɢ ɬɨɱɟɤ ɜɨɡɜɪɚɬɚ. Ƚɪɚɮɢɤ ɤɭɫɨɱɧɨ-ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ ɫɨɫɬɨɢɬ ɢɡ
ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɝɥɚɞɤɢɯ ɞɭɝ; ɬɚɤɚɹ ɤɪɢɜɚɹ ɥɢɧɢɹ ɧɚɡɵɜɚɟɬɫɹ
ɦɨɠɟɬ ɢɦɟɬɶ ɥɢɲɶ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɬɨɱɟɤ ɪɚɡɪɵɜɚ 1-ɝɨ ɪɨɞɚ. ɇɚɩɪɢɦɟɪ, ɮɭɧɤɰɢɹ,
ɝɪɚɮɢɤ ɤɨɬɨɪɨɣ ɢɡɨɛɪɚɠɟɧ ɧɚ ɪɢɫ. 8.3 ɤɭɫɨɱɧɨ-ɝɥɚɞɤɚɹ ɜ ɢɧɬɟɪɜɚɥɟ
ɦɨɠɧɨ ɪɚɡɛɢɬɶ ɧɚ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɱɚɫɬɢɱɧɵɯ ɢɧɬɟɪɜɚɥɨɜ, ɜ ɤɚɠɞɨɦ ɢɡ
()
ba,
()
xf
Ƚɪɚɮɢɤɨɦ ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ ɹɜɥɹɟɬɫɹ
ȼ ɫɢɥɭ ɞɚɧɧɵɯ ɨɩɪɟɞɟɥɟɧɢɣ ɤɭɫɨɱɧɨ-ɝɥɚɞɤɚɹ ɮɭɧɤɰɢɹ ɜ ɢɧɬɟɪɜɚɥɟ
ɧɚɡɵɜɚɟɬɫɹ
()
xf
ɧɚɡɵɜɚɟɬɫɹ
– ɝɥɚɞɤɚɹ ɮɭɧɤɰɢɹ.
ɝɥɚɞɤɨɣ
ɤɭɫɨɱɧɨ-ɝɥɚɞɤɨɣ
xf
.
()
xf
.
ɜ ɢɧɬɟɪɜɚɥɟ
ɝɥɚɞɤɚɹ ɥɢɧɢɹ
y
, ɟɫɥɢ ɜ ɷɬɨɦ ɢɧɬɟɪ-
()
ba,
ɜ ɢɧɬɟɪɜɚɥɟ
, ɧɚ ɬɚɤɨɣ ɥɢɧɢɢ ɧɟɬ ɧɢ
()
xf
ɫɯɨɞɢɬɫɹ ɢ
′
()
xf
.
, ɟɫɥɢ ɢɧɬɟɪ-
()
ba,
ɤɭɫɨɱɧɨ-ɝɥɚɞɤɨɣ.
()
.
()
ba,
ba,
,
a b
ȼ ɬɨɱɤɟ ɪɚɡɪɵɜɚ ɩɟɪɜɨɝɨ ɪɨɞɚ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɩɪɟɞɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɫɥɟɜɚ ɢ
ɫɩɪɚɜɚ, ɤɨɬɨɪɵɟ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ
ɋɮɨɪɦɭɥɢɪɭɟɦ ɛɟɡ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɨɫɧɨɜɧɭɸ ɬɟɨɪɟɦɭ ɨ ɜɨɡɦɨɠɧɨɫɬɢ ɪɚɡɥɨ-
()
ɠɟɧɢɹ ɮɭɧɤɰɢɢ
Ɍɟɨɪɟɦɚ 8.1.
ȿɫɥɢ ɮɭɧɤɰɢɹ
ɫɯɨɞɢɬɫɹ ɤ ɮɭɧɤɰɢɢ
ȼ ɬɨɱɤɚɯ ɪɚɡɪɵɜɚ ɮɭɧɤɰɢɢ
ɟɟ ɩɪɟɞɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɫɥɟɜɚ ɢ ɫɩɪɚɜɚ, ɬ. ɟ. ɤ ɡɧɚɱɟɧɢɸ
ɝɞɟ
– ɬɨɱɤɚ ɪɚɡɪɵɜɚ ɩɟɪɜɨɝɨ ɪɨɞɚ.
x
0
xf
ɜ ɪɹɞ Ɏɭɪɶɟ.
()
xf
ɤɭɫɨɱɧɨ-ɝɥɚɞɤɚɹ ɜ ɢɧɬɟɪɜɚɥɟ
()
xf
ɜɨ ɜɫɟɯ ɬɨɱɤɚɯ, ɜ ɤɨɬɨɪɵɯ ɨɧɚ ɧɟɩɪɟɪɵɜɧɚ.
0
Ɋɢɫ. 8.3
()
()
xf
ɪɹɞ ɫɯɨɞɢɬɫɹ ɤ ɫɪɟɞɧɟɦɭ ɚɪɢɮɦɟɬɢɱɟɫɤɨɦɭ
()()
2
123
0−xf
ɢ
++− xfxf
00
()
00
x
0+xf
.
()
,
, ɬɨ ɟɟ ɪɹɞ Ɏɭɪɶɟ
ππ− ,
ȼ ɨɛɟɢɯ ɝɪɚɧɢɱɧɵɯ ɬɨɱɤɚɯ ɢɧɬɟɪɜɚɥɚ ɫɭɦɦɚ ɪɹɞɚ ɪɚɜɧɚ ɫɪɟɞɧɟɦɭ ɚɪɢɮɦɟɬɢɱɟɫɤɨɦɭ ɩɪɟɞɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɮɭɧɤɰɢɢ ɩɪɢ ɫɬɪɟɦɥɟɧɢɢ ɧɟɡɚɜɢɫɢɦɨɣ ɩɟɪɟɦɟɧɧɨɣ ɤ ɷɬɢɦ ɬɨɱɤɚɦ ɢɡɧɭɬɪɢ ɢɧɬɟɪɜɚɥɚ
()()
2
()
ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɮɭɧɤɰɢɹ
()
ɡɚɦɤɧɭɬɨɦ ɢɧɬɟɪɜɚɥɟ.
ɱɬɨɛɵ ɮɭɧɤɰɢɹ
ɦɨɜ ɢ ɦɢɧɢɦɭɦɨɜ ɢ ɛɵɥɚ ɧɟɩɪɟɪɵɜɧɨɣ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ, ɛɵɬɶ ɦɨɠɟɬ, ɤɨɧɟɱɧɨɝɨ
ɱɢɫɥɚ ɬɨɱɟɤ ɪɚɡɪɵɜɚ ɩɟɪɜɨɝɨ ɪɨɞɚ (ɭɫɥɨɜɢɹ Ⱦɢɪɢɯɥɟ). ɉɪɢ ɫɨɛɥɸɞɟɧɢɢ ɷɬɢɯ
ɭɫɥɨɜɢɣ ɮɭɧɤɰɢɹ ɬɚɤɠɟ ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɪɹɞ Ɏɭɪɶɟ, ɫɯɨɞɹɳɢɣɫɹ ɜ ɬɨɱɤɚɯ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɮɭɧɤɰɢɢ ɤ ɫɚɦɨɣ ɮɭɧɤɰɢɢ, ɚ ɜ ɬɨɱɤɚɯ ɟɟ ɪɚɡɪɵɜɚ
ɨɬɧɨɫɹɬɫɹ ɤ ɨɱɟɧɶ ɲɢɪɨɤɨɦɭ ɤɥɚɫɫɭ ɮɭɧɤɰɢɣ. ȼ ɬɨ ɜɪɟɦɹ ɤɚɤ ɩɟɪɜɵɟ ɞɨɩɭɫɤɚɸɬ
ɭ ɮɭɧɤɰɢɢ ɛɟɫɤɨɧɟɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɬɨɱɟɤ ɷɤɫɬɪɟɦɭɦɨɜ, ɬɪɟɛɭɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ
ɧɟɩɪɟɪɵɜɧɨɣ ɩɟɪɜɨɣ ɩɪɨɢɡɜɨɞɧɨɣ (ɤɪɨɦɟ, ɛɵɬɶ ɦɨɠɟɬ, ɤɨɧɟɱɧɨɝɨ ɱɢɫɥɚ ɬɨɱɟɤ),
ɜɬɨɪɵɟ ɨɝɪɚɧɢɱɢɜɚɸɬ ɱɢɫɥɨ ɷɤɫɬɪɟɦɭɦɨɜ, ɧɨ ɡɚɬɨ ɧɟ ɩɪɟɞɴɹɜɥɹɸɬ ɧɢɤɚɤɢɯ ɬɪɟɛɨɜɚɧɢɣ ɤ ɫɭɳɟɫɬɜɨɜɚɧɢɸ
ɚɧɚɥɢɡɟ ɢ ɜ ɟɝɨ ɩɪɢɦɟɧɟɧɢɹɯ, ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɩɪɢɜɟɞɟɧɧɵɦ ɭɫɥɨɜɢɹɦ. ȼɦɟɫɬɟ ɠɟ
ɨɧɢ, ɛɟɡɭɫɥɨɜɧɨ, ɨɯɜɚɬɵɜɚɸɬ ɜɫɟ ɜɨɡɦɨɠɧɵɟ ɮɭɧɤɰɢɢ, ɤɚɤɢɟ ɬɨɥɶɤɨ ɨɛɵɱɧɨ ɦɨɝɭɬ ɜɫɬɪɟɬɢɬɶɫɹ, ɬɚɤ ɱɬɨ ɤɚɠɞɚɹ ɬɚɤɚɹ ɮɭɧɤɰɢɹ ɩɪɟɞɫɬɚɜɢɦɚ ɫɜɨɢɦ ɪɹɞɨɦ Ɏɭɪɶɟ.
Ɉɞɧɚɤɨ, ɞɥɹ ɫɩɪɚɜɟɞɥɢɜɨɫɬɢ ɩɪɟɞɥɨɠɟɧɢɹ ɨ ɪɚɡɥɨɠɢɦɨɫɬɢ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ
ɨɞɧɨɣ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɧɟɞɨɫɬɚɬɨɱɧɨ. ɋɭɳɟɫɬɜɭɸɬ
ɰɢɣ, ɪɹɞɵ Ɏɭɪɶɟ ɤɨɬɨɪɵɯ ɪɚɫɯɨɞɹɬɫɹ ɜ ɧɟɤɨɬɨɪɵɯ ɬɨɱɤɚɯ.
ɱɢɬɟɥɶɧɨ ɦɟɧɟɟ ɠɟɫɬɤɢɟ, ɱɟɦ ɩɪɢ ɪɚɡɥɨɠɟɧɢɢ ɜ ɫɬɟɩɟɧɧɨɣ ɪɹɞ. Ɍɚɤ ɤɚɤ, ɟɫɥɢ
ɮɭɧɤɰɢɹ ɩɪɟɞɫɬɚɜɥɟɧɚ ɪɹɞɨɦ Ɍɟɣɥɨɪɚ, ɬɨ ɨɧɚ ɜɨ ɜɫɟɦ ɢɧɬɟɪɜɚɥɟ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
ɧɟ ɬɨɥɶɤɨ ɧɟɩɪɟɪɵɜɧɚ, ɧɨ ɢ
ɥɨɠɟɧɢɹ ɠɟ ɮɭɧɤɰɢɢ ɜ ɪɹɞ Ɏɭɪɶɟ ɷɬɨɝɨ ɧɟ ɬɪɟɛɭɟɬɫɹ; ɧɭɠɧɨ ɬɨɥɶɤɨ, ɱɬɨɛɵ ɫɭɳɟɫɬɜɨɜɚɥɚ ɢ ɛɵɥɚ ɧɟɩɪɟɪɵɜɧɨɣ (ɢ ɬɨ ɧɟ ɜɫɸɞɭ) ɩɟɪɜɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɮɭɧɤɰɢɢ.
ɪɨɲɟɟ ɟɟ ɩɪɢɛɥɢɠɟɧɢɟ, ɜɡɹɜ ɤɨɧɟɱɧɭɸ ɫɭɦɦɭ ɱɥɟɧɨɜ ɪɹɞɚ
ɪɹɞɨɦ Ɍɟɣɥɨɪɚ ɷɬɚ ɤɨɧɟɱɧɚɹ ɫɭɦɦɚ ɱɥɟɧɨɜ ɧɚɡɵɜɚɟɬɫɹ
ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ, ɬɨ ɪɚɡɥɨɠɟɧɢɟ ɟɟ ɜ ɪɹɞ Ɏɭɪɶɟ ɫɩɪɚɜɟɞɥɢɜɨ ɜɨ ɜɫɟɦ
ππ− ,
ɍɫɥɨɜɢɹ ɨɫɧɨɜɧɨɣ ɬɟɨɪɟɦɵ ɦɨɝɭɬ ɛɵɬɶ ɧɟɫɤɨɥɶɤɨ ɢɧɵɦɢ: ɦɨɠɧɨ ɬɪɟɛɨɜɚɬɶ,
()
xf
ɢɦɟɥɚ ɜ ɢɧɬɟɪɜɚɥɟ
()()
2
ɍɫɥɨɜɢɹ, ɧɚɤɥɚɞɵɜɚɟɦɵɟ ɧɚ ɮɭɧɤɰɢɸ ɩɪɢ ɪɚɡɥɨɠɟɧɢɢ ɟɟ ɜ ɪɹɞ Ɏɭɪɶɟ, ɡɧɚ-
ȿɫɥɢ ɮɭɧɤɰɢɹ ɪɚɡɥɚɝɚɟɬɫɹ ɜ ɪɹɞ Ɏɭɪɶɟ, ɬɨ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɞɨɫɬɚɬɨɱɧɨ ɯɨ-
00
++− xfxf
00
. ɍɫɥɨɜɢɹ ɨɫɧɨɜɧɨɣ ɬɟɨɪɟɦɵ, ɪɚɜɧɨ ɤɚɤ ɢ ɭɫɥɨɜɢɹ Ⱦɢɪɢɯɥɟ,
ɩɪɨɢɡɜɨɞɧɨɣ. Ʌɸɛɚɹ ɢɡ ɮɭɧɤɰɢɣ, ɭɩɨɬɪɟɛɥɹɟɦɵɯ ɜ
() ()
xf
xf
ɝɥɚɞɤɚɹ ɢ ɟɟ ɡɧɚɱɟɧɢɹ ɧɚ ɤɨɧɰɚɯ ɢɧɬɟɪɜɚɥɚ
ɛɟɫɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɪɚɡ ɞɢɮɮɟɪɟɧɰɢɪɭɟɦɚ. Ⱦɥɹ ɪɚɡ-
k
a
0
¦
2
n
=
1
00 −π++π− ff
. Ŷ
()
ɥɢɲɶ ɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɦɚɤɫɢɦɭ-
ππ− ,
ɩɪɢɦɟɪɵ ɧɟɩɪɟɪɵɜɧɵɯ ɮɭɧɤ-
ɦɧɨɝɨɱɥɟɧɨɦ Ɏɭɪɶɟ:
++≈
nxbnxa
sincos
nn
– ɤ ɡɧɚɱɟɧɢɸ
()
x
0
Ɏɭɪɶɟ. ɉɨ ɚɧɚɥɨɝɢɢ ɫ
.
124
Ɂɞɟɫɶ ɬɚɤɠɟ ɜɫɬɚɟɬ ɜɨɩɪɨɫ ɨ ɜɟɥɢɱɢɧɟ ɨɲɢɛɤɢ ɷɬɨɝɨ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ. Ɉɞɧɚɤɨ ɷɬɚ ɡɚɞɚɱɚ ɛɨɥɟɟ ɫɥɨɠɧɚɹ, ɱɟɦ ɩɪɢ ɡɚɦɟɧɟ ɮɭɧɤɰɢɢ ɦɧɨɝɨɱɥɟɧɨɦ
Ɍɟɣɥɨɪɚ, ɢ ɦɵ ɤɚɫɚɬɶɫɹ ɟɟ ɧɟ ɛɭɞɟɦ.
ɉɪɢɦɟɪ 8.1.
Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɸ
()
xf
, ɨɩɪɟɞɟɥɟɧɧɭɸ ɜ ɢɧɬɟɪɜɚɥɟ
()
ππ− ,
ɬɚɤ:
≤<π−
,0 ɩɪɢ
()
=
xf
®
¯
xax
π<<
,0 ɩɪɢ
xbx
ɝɞɟ a ɢ b – ɩɨɫɬɨɹɧɧɵɟ.
Ɋɟɲɟɧɢɟ.
()
bxy =
xf
ɹɜɥɹɟɬɫɹ ɥɨɦɚɧɚɹ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɞɜɭɯ ɨɬɪɟɡɤɨɜ ɩɪɹɦɵɯ:
(ɩɪɢ
0<a
ɢ
0>b
ɪɢɫ. 8.4). Ɏɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ
y
Ƚɪɚɮɢɤɨɦ
axy =
ɢ
ɬɟɨɪɟɦɵ 8.1.
,
ȼɵɱɢɫɥɢɦ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ
ɉɪɢ
0≠n
ɢɦɟɟɦ
ππ
111
a f x nxdx ax nxdx bx nxdx
==+=
n
πππ
§·§·
a x nx nx b x nx nx
=+++=
¨¸¨¸
¨¸¨¸
ππ
©¹©¹
abab
=− + −= −−
nn n
ππ π
cos cos cos
()
³³³
ππ
−−
00
sin cos sin cos
nn
−−
1cos cos 1 1 1 ,
()() ()
22 2
22
nn
ππ
nn
ππ
0
Ɋɢɫ. 8.4
0
125
x
()
xf
ɩɨ ɮɨɪɦɭɥɚɦ (8.7).
0
ππ
0
−
ªº
¬¼
o
2
(ɡɞɟɫɶ ɢ ɞɚɥɟɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ ɱɚɫɬɹɦ), ɬ. ɟ.
f
−
ba
=
a
π
,0,2
−
ba
==⋅
aa
321
2
3
⋅π
=⋅
a
4
ɉɪɢ
0=n ɢɦɟɟɦ
0
=
0
π
π−
π
11
+
³³
π
0
−
ab
.
π
=
dxbxdxaxa
2
Ⱦɚɥɟɟ
ππ
111
b f x nxdx ax nxdx bx nxdx
==+=
n
πππ
−−
sin sin sin
()
³³³
ππ
0
0
...,0,2
.
§·§·
a x nx nx b x nx nx
=− + +− + =
¨¸¨¸
¨¸¨¸
ππ
©¹©¹
00
cos sin cos sin
nn
ππ
−−
ππ
−+
abab
=−=−
ππ
ππ
cos cos 1 .
nnn
ɬ. ɟ.
ba
+
b
=
b
−=
,
1
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɹɞ Ɏɭɪɶɟ ɢɦɟɟɬ ɜɢɞ
ba ab ab ab
−− + +
xxxx
=+⋅+ − +
()
ab ab ab
§·
+⋅ + − +
¨¸
©¹
§·
π
¨¸
412
−++
π
π
©¹
2
cos3 sin3 sin 4 ...
2
3
ππ
22
nn
nn
ba
+
b
,
2
ba
+
=
3
00
n
+
1
()
ba
b
,
4321
+
−=
4
...,
.
2cos sin sin2
xx x
34
126
ɢɥɢ
ȼ ɢɧɬɟɪɜɚɥɟ
ɪɚɜɟɧ
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ
(ɪɢɫ. 8.5).
2
ab
−
ba
−
=+ + ++
fx x x
()
()
4
++⋅ − + −
ab x x x
()
ɪɹɞ ɩɪɟɞɫɬɚɜɥɹɟɬ ɮɭɧɤɰɢɸ
ππ− ,
()()
()
π
§·
sin sin 2 sin3 ... .
¨¸
©¹
()
=
xf
®
¯
§·
cos cos3 ...
¨¸
π
©¹
11
23
00 abbaff −
−π++π−
=
0,1 == ba
, ɬɨ
y
1
2
3
()
xf
, ɚ ɜ ɬɨɱɤɚɯ
π+π−
π=
222
≤<π−
,0ɩɪɢ
xx
π><
,0ɩɪɢ0
x
π±=x
ɨɧ
.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɥɭɱɚɟɦ ɪɹɞ
π
2
()
ɗɬɨɬ ɪɹɞ ɜ ɢɧɬɟɪɜɚɥɟ
ɉɪɢ π±=x ɫɭɦɦɚ ɟɝɨ ɪɚɜɧɚ
§
+
−= xxxxxxxf
cos
¨
π
4
©
1
3
()
0
Ɋɢɫ. 8.5
1
3cos
22
5
ɪɚɜɟɧ x, ɚ ɜ ɢɧɬɟɪɜɚɥɟ
0,π−
π
. Ź
−
2
+++
127
x
·
§
¸
¨
©
¹
1
sin...5cos
2
()
1
2sin
3
ɫɯɨɞɢɬɫɹ ɤ ɧɭɥɸ.
π,0
−+−+
·
....3sin
¸
¹
8.2. Ɋɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɏɭɪɶɟ ɱɟɬɧɵɯ
ɢ ɧɟɱɟɬɧɵɯ ɮɭɧɤɰɢɣ
()
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɪɚɡɥɚɝɚɟɦɚɹ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɹ
ɰɢɢ
()
ɱɟɬɧɵɯ ɮɭɧɤɰɢɣ ɩɨ ɢɧɬɟɪɜɚɥɭ
ɛɭɞɭɬ ɧɟɱɟɬɧɵɦɢ, ɢ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
kxxf sin
–ʌ, ʌ
()
, ɫɢɦɦɟɬɪɢɱɧɨɦɭ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɚɱɚɥɚ ɤɨ-
ɨɪɞɢɧɚɬ, ɨɤɚɠɭɬɫɹ ɪɚɜɧɵɦɢ ɧɭɥɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɪɹɞ
Ɏɭɪɶɟ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɨɞɧɢɯ ɤɨɫɢɧɭɫɨɜ:
ɝɞɟ
1
() ()
π
π−
()
ɢɛɨ
()
– ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ.
nxxf cos
Ⱦɨɩɭɫɬɢɦ ɬɟɩɟɪɶ, ɱɬɨ
=
n
ɧɟɱɟɬɧɵɦɢ ɮɭɧɤɰɢɹɦɢ, ɢ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
∞
a
0
+
n
¦
2
1
n
=
cos
xf
– ɧɟɱɟɬɧɚɹ ɮɭɧɤɰɢɹ. Ɍɨɝɞɚ
, (8.8)
cos
nxa
ππ
2
=
³³
π
0
ɨɤɚɡɵɜɚɸɬɫɹ ɧɭɥɹɦɢ. ɋɥɟɞɨɜɚ-
a
k
ɬɟɥɶɧɨ, ɧɟɱɟɬɧɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɪɹɞ Ɏɭɪɶɟ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɨɞɧɢɯ ɫɢɧɭɫɨɜ:
¦
n
∞
, (8.9)
sin
nxb
n
1
=
ɝɞɟ
ɢɛɨ
()
1
=
n
π
π−
– ɱɟɬɧɚɹ ɮɭɧɤɰɢɹ.
nxxf sin
() ()
sin
ππ
2
=
³³
π
0
ɉɪɢɦɟɪ 8.2.
Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɸ
()
Ɋɟɲɟɧɢɟ.
Ƚɪɚɮɢɤ ɷɬɨɣ ɮɭɧɤɰɢɢ, ɩɟɪɢɨɞɢɱɟɫɤɢ ɩɪɨɞɨɥɠɟɧɧɵɣ ɧɚ ɜɫɸ ɱɢɫɥɨɜɭɸ ɨɫɶ,
ɢɡɨɛɪɚɠɟɧ ɧɚ ɪɢɫ. 8.6. Ɏɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ 8.1. ɉɨɫɤɨɥɶɤɭ
()
ɮɭɧɤɰɢɹ
xf
ɱɟɬɧɚɹ, ɬɨ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
xf
– ɱɟɬɧɚɹ. Ɍɨɝɞɚ ɮɭɧɤ-
, ɤɚɤ ɢɧɬɟɝɪɚɥɵ ɨɬ ɧɟ-
b
k
cos
sin
π<<π−= xxxf ,
.
.
0=nb
,
dxnxxfdxnxxfa
kxxf cos
ɛɭɞɭɬ
()
,
dxnxxfdxnxxfb
128
ɉɨ ɮɨɪɦɭɥɚɦ (8.8) ɢɦɟɟɦ:
π
=
0
π
0
ɂɧɬɟɝɪɢɪɭɹ ɩɨ ɱɚɫɬɹɦ, ɩɨɥɭɱɢɦ
ª
«
a
=
n
π
«
¬
π
nxx
n
0
π
2
22
x
dxxa
=
2
π
π
nx
−
³
n
0
dx
,
0
º
2sinsin2
»
=
2
π
n
»
¼
y
n
cos
π
2
cos
=π=
³³
π
0
π
=
nx
0
dxnxxa
.
2
()
2
π
n
.1cos
−π
n
Ɉɬɫɸɞɚ
ɂɬɚɤ,
ɉɪɢ
ɂɦɟɟɦ
ɨɬɤɭɞɚ
0
4
−=
π
π
()
= xxxxf
2
ɩɨɥɭɱɚɟɬɫɹ ɢɧɬɟɪɟɫɧɵɣ ɱɢɫɥɨɜɨɣ ɪɹɞ ɞɥɹ ɱɢɫɥɚ π.
0=x
Ɋɢɫ. 8.6
4
−==
,0,
321
2
π
3
4
§
−
cos
¨
π
©
π
=
0
2
2
π
8
1
3
4
§
−
1
¨
π
3
©
1
1
3
,0,
aaaaa
1
3cos
22
5
1
1
+++
...
22
5
1
.
...
+++=
22
5
4
−==
5
2
π
+++
ɢ ɬ. ɞ.
·
¸
54
¹
·
,
¸
¹
129
x
....5cos
ɂɡ ɷɬɨɣ ɮɨɪɦɭɥɵ ɩɨɥɭɱɚɸɬɫɹ ɧɟɤɨɬɨɪɵɟ ɞɪɭɝɢɟ ɱɢɫɥɨɜɵɟ ɪɹɞɵ.
ɉɭɫɬɶ
1
1
2
222
4
3
1
1
1
1
1
1,...
22
5
3
2
π
...
,
=+++=σ++++=σ
8
2
2
6
4
1
1
1
§
1
...
¨
4
©
1
1
2
·
.
+++=++++=σ ...
¸
22222
3
¹
1
1
Ɍɨɝɞɚ
1
, σ+σ=σ=σσ+σ=σ
1
4
()
4
,
21221
ɢɥɢ
2
3 σ=σ
, ɬ. ɟ.
12
π
.
=σ
2
24
Ɂɧɚɱɢɬ,
π
=σ
=
+
6248
, ɬ. ɟ.
222
π
π
2
π
1
6
1
1
3
2
1
1
4
. Ź
...
+++++=
2222
5
ɉɪɢɦɟɪ 8.3.
()
Ɋɚɡɥɨɠɢɬɶ ɜ ɪɹɞ Ɏɭɪɶɟ ɮɭɧɤɰɢɸ
xf
, ɨɩɪɟɞɟɥɟɧɧɭɸ ɜ ɢɧɬɟɪɜɚɥɟ
ɬɚɤ:
<<π−−
,0 ɩɪɢ1
()
=
xf
®
¯
x
π<<
.0 ɩɪɢ1
x
Ɋɟɲɟɧɢɟ.
Ƚɪɚɮɢɤ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ, ɩɟɪɢɨɞɢɱɟɫɤɢ ɩɪɨɞɨɥɠɟɧɧɵɣ ɧɚ ɜɫɸ ɨɫɶ
ɪɚɠɟɧ ɧɚ ɪɢɫ. 8.7.
ɗɬɚ ɮɭɧɤɰɢɹ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɫɥɨɜɢɹɦ ɬɟɨɪɟɦɵ 8.1. ɉɨɫɤɨɥɶɤɭ ɨɧɚ ɧɟɱɟɬɧɚɹ,
ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɮɨɪɦɭɥɚɦɢ (8.9) ɢɦɟɟɦ
0=na
π
2
=
n
sin1
³
π
0
dxnxb
2
cos
−=⋅
π
xn
n
2
π
0
()
−=
π
n
,
1cos
−π
n
()
ππ− ,
Ox , ɢɡɨɛ-
130
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