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ɉɈɇəɌɂȿ Ɉ ȼɕɑɂɋɅɂɌȿɅɖɇɈɃ ȽȿɈɆȿɌɊɂɂ
ɩɨɜɟɪɧɭɜ ɧɚɲɭ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ ɧɚ ɭɝɨɥ ȕ, ɩɪɢɞɟɦ ɤ ɫɥɟɞɭɸɳɟɣ ɫɢɫɬɟɦɟ
ɤɨɨɪɞɢɧɚɬ:
ɢ, ɩɪɢɪɚɜɧɹɜ ɤ ɧɭɥɸ ɫɭɦɦɭ
c
®
c
¯
n
¦
1
i
ȕyȕxx
sincos
ȕyȕxy
cossin
,
cc
yx
, ɧɚɣɞɟɦ ɭɝɨɥ ȕ ɢɡ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɝɨ
ii
ɭɪɚɜɧɟɧɢɹ:
n
cc
ii
1
i
n
22
ii
1
i
n
¦¦¦
i
22
.)(sincos)sin(cos0
yxȕȕyxȕȕyx
ii
1
ɉɭɫɬɶ ɬɟɩɟɪɶ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɬɚɤɨɜɚ, ɱɬɨ ɜɵɩɨɥɧɹɸɬɫɹ ɭɫɥɨɜɢɹ:
n
n
i
i
011
yx
¦¦
i
ɢ
¦
i
ni
yx
ii
,01
(24)
ɢ ɧɚɲɟɣ ɡɚɞɚɱɟɣ ɛɭɞɟɬ ɢɫɫɥɟɞɨɜɚɬɶ ɧɚ ɷɤɫɬɪɟɦɭɦ ɮɭɧɤɰɢɸ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ
S
(Į, 0).
2
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ
ɢɥɢ
i
n
2
1
i
n
2
sin
D
i
1
i
n
n
1
n
§
¨
1
i
©
2
2
!
yx
, ɬɨɝɞɚ ɢɡ ɪɚɜɟɧɫɬɜ (24) ɢɦɟɟɦ
¦¦
i
i
i
1
2
ii
n
·
2
22
yxy
¸
¦¦¦
i
i
1
i
¹
n
2
D
2
i
1
i
n
22
i
1
i
n
2
,)0,(
dd
xSy
¦¦
i
1
i
n
22
xyxyS
sincos)sincos()0,(
DDDDD
¦¦¦
i
1
i
ʌ
ɝɞɟ ɡɧɚɤɢ ɪɚɜɟɧɫɬɜɚ ɞɨɫɬɢɝɚɸɬɫɹ ɩɪɢ
ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɹɜɥɹɟɬɫɹ ɨɫɶ Ɉ
ɝɢɱɧɨ.
ȿɫɥɢ
n
n
2
yx
¦¦
i
i
i
i
1
1
ɯ. ɋɥɭɱɚɣ
2
, ɬɨ ɪɟɲɟɧɢɣ ɛɭɞɟɬ ɛɟɫɤɨɧɟɱɧɨ ɦɧɨɝɨ, ɬɨ ɟɫɬɶ ɪɟɲɟ-
Į = 0 ɢ
Į
. Ɍɨ ɟɫɬɶ ɢɫɤɨɦɨɣ ɩɪɹɦɨɣ
2
n
n
i
1
2
2
yx
ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɚɧɚɥɨ-
¦¦
i
i
i
1
ɧɢɟɦ ɧɚɲɟɣ ɡɚɞɚɱɢ ɛɭɞɟɬ ɥɸɛɚɹ ɩɪɹɦɚɹ (ɩɭɱɨɤ ɩɪɹɦɵɯ), ɩɪɨɯɨɞɹɳɚɹ ɱɟɪɟɡ
ɰɟɧɬɪ ɦɚɫɫ ɫɢɫɬɟɦɵ ɬɨɱɟɤ
Ⱥ
, Ⱥ2, …, Ⱥn ɫ ɟɞɢɧɢɱɧɵɦɢ ɦɚɫɫɚɦɢ.
1
81
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
E
S
E
ȿɫɥɢ ɠɟ ɨɞɧɨ ɢɡ ɱɢɫɟɥ D ɢɥɢ E ɧɟ ɪɚɜɧɨ ɧɭɥɸ, ɬɨ ɩɪɢɯɨɞɢɦ ɤ ɞɜɭɦ ɪɟɲɟɧɢɹɦ:
ɢ
1
D
arctg
D
2
2
.
2
D
1
D
arctg
1
2
ȼ ɦɟɯɚɧɢɤɟ ɷɬɢ ɧɚɣɞɟɧɧɵɟ ɩɪɹɦɵɟ (
ɨɫɹɦɢ ɢɧɟɪɰɢɢ ɫɢɫɬɟɦɵ ɦɚɬɟɪɢɚɥɶɧɵɯ ɬɨɱɟɤ
ɯ = 0 ɢ ɭ = 0) ɧɚɡɵɜɚɸɬɫɹ ɝɥɚɜɧɵɦɢ
Ⱥ
, Ⱥ2, …, Ⱥn ɫ ɟɞɢɧɢɱɧɵɦɢ
1
ɦɚɫɫɚɦɢ.
ɂɬɚɤ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɬɟɨɪɟɦɟ.
Ɍɟɨɪɟɦɚ 25 (ɡɚɞɚɱɚ ȼȽ5). Ⱦɥɹ ɥɸɛɵɯ Ⱥ
ɧɚɢɥɭɱɲɚɹ ɜ ɫɦɵɫɥɟ ɤɜɚɞɪɚɬɢɱɟɫɤɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ (ɬɨ ɟɫɬɶ ɜ
, …, Ⱥn, (n > 2) ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ
1
l
-ɦɟɬɪɢɤɟ)
2
ɩɪɹɦɚɹ ɫɭɳɟɫɬɜɭɟɬ, ɢ ɟɞɢɧɫɬɜɟɧɧɚɹ, ɢɥɢ ɬɚɤɢɯ ɩɪɹɦɵɯ ɛɟɫɤɨɧɟɱɧɨ ɦɧɨɝɨ,
ɢ ɜɫɟ ɨɧɢ ɩɪɨɯɨɞɹɬ ɱɟɪɟɡ ɰɟɧɬɪ ɦɚɫɫ ɷɬɨɣ ɫɢɫɬɟɦɵ ɬɨɱɟɤ.
l2 l
l1
1
1
1
.5
0
0 0.5 1
0.5
0
0 0.5 1
0.5
0
0 0.5 1
Ⱦɟɦɨɧɫɬɪɚɰɢɹ ɪɟɲɟɧɢɣ ɡɚɞɚɱ ȼȽ5, ȼȽ6 ɢ ȼȽ4.
ɉɨɫɥɟɞɧɹɹ ɬɟɨɪɟɦɚ ɢ ɪɚɫɫɭɠɞɟɧɢɹ, ɫɜɹɡɚɧɧɵɟ ɫ ɧɟɣ, ɩɪɢɜɨɞɢɬ ɧɚɫ
ɤ ɫɥɟɞɭɸɳɟɣ ɡɚɞɚɱɟ.
Ɂɚɞɚɱɚ ȼȽ7. ɉɭɫɬɶ ɢɦɟɟɬɫɹ n ɬɨɱɟɤ Ⱥ
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɚɤɭɸ ɬɨɱɤɭ (
Ⱥ
, …, Ⱥn ɞɨ ɥɸɛɨɣ ɩɪɹɦɨɣ l ɛɭɞɟɬ ɩɨɫɬɨɹɧɧɨɣ (ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ l) ɜɟɥɢɱɢɧɨɣ.
2
ȼɟɪɧɚ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ.
ɯ
, ɭ0), ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɪɚɫɫɬɨɹɧɢɣ ɨɬ ɬɨɱɟɤ Ⱥ1,
0
, Ⱥ2, …, Ⱥn (n > 2) ɧɚ ɩɥɨɫɤɨɫɬɢ.
1
Ɍɟɨɪɟɦɚ 26 (ɡɚɞɚɱɚ ȼȽ7). Ⱦɥɹ ɥɸɛɨɣ ɫɢɫɬɟɦɵ n ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ ɫɭɳɟ-
ɫɬɜɭɟɬ ɪɨɜɧɨ ɞɜɟ ɬɨɱɤɢ, ɧɟɨɛɹɡɚɬɟɥɶɧɨ ɪɚɡɥɢɱɧɵɟ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɟ ɭɫɥɨɜɢɸ
ɡɚɞɚɱɢ, ɩɪɢ ɷɬɨɦ ɨɧɢ ɫɢɦɦɟɬɪɢɱɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɦɚɫɫ ɷɬɨɣ ɫɢɫɬɟɦɵ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɭɫɬɶ Ⱥ
, yi), i = 1, …, n, – ɬɨɱɤɢ ɩɥɨɫɤɨɫɬɢ. ȼ ɜɢɞɭ
i(xi
ɩɨɫɬɚɧɨɜɤɢ ɡɚɞɚɱɢ ɛɟɡ ɩɨɬɟɪɢ ɨɛɳɧɨɫɬɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɜɵɛɪɚɧɚ ɬɚɤ, ɱɬɨ ɢɦɟɸɬ ɦɟɫɬɨ ɪɚɜɟɧɫɬɜɚ (24).
Ⱦɚɥɟɟ, ɩɭɫɬɶ
ɧɢɣ ɨɬ ɬɨɱɟɤ
Ɋ ɪɚɜɧɨ
Ɋ(ɯ
, ɭ0) – ɢɫɤɨɦɚɹ ɬɨɱɤɚ, ɬɨɝɞɚ ɫɭɦɦɭ ɤɜɚɞɪɚɬɨɜ ɪɚɫɫɬɨɹ-
Ⱥ
i(xi
0
, yi), i = 1, …, n, ɞɨ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɪɹɦɨɣ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ
¦
n
i
1
ii
2
yyxxS
.)sin)(cos)(()(
GGG
00
Ɉɬɤɪɵɜɚɹ ɫɤɨɛɤɢ ɢ ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɚɦɢ:
82
ɉɈɇəɌɂȿ Ɉ ȼɕɑɂɋɅɂɌȿɅɖɇɈɃ ȽȿɈɆȿɌɊɂɂ
2sinįcosį = sin2į, 2cos
2
į = 1 + cos2į, ɢ 2sin2į = 1 – cos2į, ɩɨɥɭɱɚɟɦ:
S(į) = S
+ S1(į),
0
S
– ɜɟɥɢɱɢɧɚ, ɧɟ ɡɚɜɢɫɹɳɚɹ ɨɬ į:
ɝɞɟ
0
n
1
¦
2
1
i
2
00
ii
2
yyxxS
)()(
, (25)
0
ɚ
S
(į) – ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɣ ɩɨɥɢɧɨɦ Fcos2į + Gsin2į ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ:
1
n
2
¦¦
0
i
1
.))((
yyxxG
ii
00
i
·
2
)()(
yyxxF
¸
0
¹
(26)
¦
n
1
§
¨
i
2
i
1
©
n
i
1
ɉɨ ɭɫɥɨɜɢɸ ɡɚɞɚɱɢ ɢɫɤɨɦɚɹ ɬɨɱɤɚ
ɮɭɧɤɰɢɹ
ɢ sin2
S(į) ɧɟ ɡɚɜɢɫɟɥɚ ɨɬ į. ȼɜɢɞɭ ɧɟɡɚɜɢɫɢɦɨɫɬɢ ɮɭɧɤɰɢɣ cos2į
į ɜɟɥɢɱɢɧɚ S
(į) ɛɭɞɟɬ ɪɚɜɧɨɣ ɧɭɥɸ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ F ɢ G
1
Ɋ(ɯ
) ɞɨɥɠɧɚ ɛɵɬɶ ɬɚɤɨɣ, ɱɬɨɛɵ
0,ɭ0
ɪɚɜɧɵ ɧɭɥɸ. Ɍɨ ɟɫɬɶ ɩɪɢɯɨɞɢɦ ɤ ɫɢɫɬɟɦɟ ɢɡ ɞɜɭɯ ɭɪɚɜɧɟɧɢɣ ɫ ɞɜɭɦɹ ɧɟɢɡ-
ɯ
ɜɟɫɬɧɵɦɢ
ɢ ɭ0:
0
n
i
°
0
F
®
0
G
¯
ɢɥɢ
°
i
1
®
n
°
¦
°
i
1
¯
n
2
¦¦
0
i
1
yyxx
ii
2
yyxx
)()(
i
0
(27)
.0))((
00
ɉɨɥɶɡɭɹɫɶ ɪɚɜɟɧɫɬɜɚɦɢ (24), ɩɪɢɯɨɞɢɦ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ:
n
2
°
i
®
i
1
°
yx
00
¯
n
¦¦
0
i
1
22
i
2
nyynxx
0
(28)
.0
Ɉɬɫɸɞɚ ɢɦɟɟɦ:
1.
ȿɫɥɢ
i
(0,0) – ɰɟɧɬɪ ɦɚɫɫ ɫɢɫɬɟɦɵ {
n
n
i
1
2
2
yx
, ɬɨ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɛɭɞɟɬ ɟɞɢɧɫɬɜɟɧɧɚɹ ɬɨɱɤɚ
¦¦
i
i
1
Ⱥ
, Ⱥ2, …, Ⱥn}.
1
83
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ȿɫɥɢ
§
¨
P
2,1
¨
©
n
i
n
1
¦
n
1
i
n
2
i
1
r
xy
2
yx
, ɬɨ ɭ0 = 0, ɢ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɛɭɞɭɬ ɬɨɱɤɢ
¦¦
i
i
1
·
22
¸
ɢɥɢ
0,)(
ii
¸
¹
§
1
¨
P
,0
2,1
¨
©
¦
n
ni
1
·
22
¸
– ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ.
r
yx
)(
ii
¸
¹
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɷɬɢ ɬɨɱɤɢ ɫɢɦɦɟɬɪɢɱɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ (0,0).
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
0.5
0
0.5
0.5 0 0.5
10
0
10
10 0 10
4
2
0
2
4
4 2 0 2 4
Ⱦɟɦɨɧɫɬɪɚɰɢɹ ɪɟɲɟɧɢɣ ɡɚɞɚɱɢ ȼȽ7 ɞɥɹ 19 ɢ ɞɥɹ 3 ɬɨɱɟɤ.
ɇɚɣɞɟɧɧɵɟ ɬɨɱɤɢ ɞɥɹ
n = 3 ɧɟ ɜɯɨɞɹɬ ɜ ɷɧɰɢɤɥɨɩɟɞɢɸ ɡɚɦɟɱɚɬɟɥɶɧɵɯ
ɥɢɧɢɣ ɢ ɬɨɱɟɤ ɬɪɟɭɝɨɥɶɧɢɤɚ, ɧɨ ɦɨɝɭɬ ɩɪɟɬɟɧɞɨɜɚɬɶ ɧɚ ɷɬɨ.
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɬɨɦ, ɤɚɤ ɦɨɝɭɬ ɪɚɫɩɨɥɚɝɚɬɶɫɹ ɬɨɱɤɢ
Ⱥ
, Ⱥ2, …, Ⱥn
1
ɧɚ ɩɥɨɫɤɨɫɬɢ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɚ. Ⱦɥɹ ɹɫɧɨɫɬɢ ɢɡɥɨɠɟɧɢɹ ɜ ɞɚɥɶɧɟɣɲɟɦ
ɛɭɞɟɦ ɩɨɥɶɡɨɜɚɬɶɫɹ ɨɛɨɡɧɚɱɟɧɢɹɦɢ, ɩɪɢɧɹɬɵɦɢ ɜ ɥɢɧɟɣɧɨɣ ɚɥɝɟɛɪɟ.
ɇɚɡɨɜɟɦ ɷɬɭ ɡɚɞɚɱɭ ɤɚɤ «ɡɚɞɚɱɚ ȼȽ8» ɢ ɩɪɢɜɟɞɟɦ ɫɥɟɞɭɸɳɭɸ ɩɨɥɟɡɧɭɸ
ɥɟɦɦɭ, ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɤɨɬɨɪɨɣ ɫɥɟɞɭɟɬ ɢɡ ɪɚɜɟɧɫɬɜ (27) ɢ ɞɪɭɝɢɯ ɫɨɨɛɪɚɠɟɧɢɣ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɬɟɨɪɟɦɵ 26:
Ʌɟɦɦɚ. Ɍɨɱɤɚ Ɋ(ɯ
Ⱥ
, ɭ2), …, Ⱥn(ɯn, ɭn) ɜ ɬɨɦ ɢ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ ɜɟɤɬɨɪɵ
2(ɯ2
yy
§
¨
¨
ɢ
g
¨
¨
¨
©
·
01
¸
yy
¸
02
ɨɪɬɨɝɨɧɚɥɶɧɵ ɢ ɢɦɟɸɬ ɪɚɜɧɵɟ ɞɥɢɧɵ.
¸
...
¸
¸
yy
¹
0
n
, ɭ0) ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɞɥɹ ɬɨɱɤɢ Ⱥ1(ɯ1, ɭ1),
0
xx
§
¨
xx
¨
f
¨
...
¨
¨
xx
©
n
·
01
¸
¸
02
¸
¸
¸
¹
0
ɂɡ ɥɟɦɦɵ ɫɥɟɞɭɟɬ, ɱɬɨ ɟɫɥɢ ɬɨɱɤɢ
Ⱥ
, Ⱥ2, …, Ⱥn ɹɜɥɹɸɬɫɹ ɜɟɪɲɢɧɚɦɢ
1
ɩɪɚɜɢɥɶɧɨɝɨ ɦɧɨɝɨɭɝɨɥɶɧɢɤɚ, ɬɨ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɛɭɞɟɬ ɰɟɧɬɪ ɦɚɫɫ ɷɬɢɯ ɬɨɱɟɤ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɛɟɡ ɩɨɬɟɪɢ ɨɛɳɧɨɫɬɢ, ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɬɨɱɤɢ
Ⱥ
, Ⱥ2, …,
1
84
ɉɈɇəɌɂȿ Ɉ ȼɕɑɂɋɅɂɌȿɅɖɇɈɃ ȽȿɈɆȿɌɊɂɂ
Ⱥ
ɢɦɟɸɬ ɤɨɨɪɞɢɧɚɬɵ
n
1
n
ɤɚɤ ɩɪɢ
n > 2
e
0
j
§
A
cos
¨
1
j
ij
S
4
n
©
1
n
4
§
cos
¨
n
0
j
©
·
§
¸
¨
n
¹
©
1
n
j
·
sin
i
¸
¦¦¦
0
j
¹
j
2
j
SS
2
·
·
§
sin,
¸
¸
¨
n
¹
©
¹
SS
j
4
·
§
¨
©
0
(i – ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ), ɬɨ
¸
n
¹
1...,,1,0,
nj
. Ɍɨɝɞɚ, ɬɚɤ
1
n
2
j
·
§
2
cos
0
j
¸
¨
n
¹
©
1
n
4
jn
·
§
cos
212
0
j
¸
¨
n
¹
©
1
n
2
jn
·
§
2
sin,
2
0
j
¸
¨
n
¹
©
1
n
4
jn
SSSS
§
cos
¨
¦¦¦¦
212
n
0
j
©
ɢ
1
n
j
j
2
§
·
§
cos
0
sin
¨
¸
¨
n
©
n
©
¹
1
n
SSS
j
12
·
¸
¦¦
2
¹
j
4
·
§
sin
¨
©
0
j
.0
¸
n
¹
Ɉɬɫɸɞɚ ɢ ɥɟɦɦɵ ɫɥɟɞɭɟɬ, ɱɬɨ
ȼ ɫɥɭɱɚɟ
n = 3 ɜɟɪɧɨ ɢ ɨɛɪɚɬɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ.
Ɋ(0,0) – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ.
Ɍɟɨɪɟɦɚ 27 (ɡɚɞɚɱɚ ȼȽ8). ȿɫɥɢ ɬɨɱɤɚ Ɋ – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ȼȽ7 ɞɥɹ ɬɨɱɟɤ
Ⱥ, ȼ, ɋ ɫɨɜɩɚɞɚɟɬ ɫ ɰɟɧɬɪɨɦ ɢɯ ɦɚɫɫ, ɬɨ ɬɪɟɭɝɨɥɶɧɢɤ Ⱥȼɋ ɩɪɚɜɢɥɶɧɵɣ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ȼɜɟɞɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ Ɉɯɭ ɬɚɤ, ɱɬɨɛɵ ɬɨɱɤɚ
Ɋ(0, 0) ɛɵɥɚ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ɞɥɹ ɬɨɱɟɤ Ⱥ, ȼ, ɋ. ɉɭɫɬɶ Ⱥ(ɯ
ɢ
ɋ(ɯ
, ɭ3) – ɜɟɪɲɢɧɵ ɬɪɟɭɝɨɥɶɧɢɤɚ Ⱥȼɋ.
3
Ɍɚɤ ɤɚɤ
y
§
1
¨
y
g
¨
2
¨
y
3
©
Ɍɚɤ ɤɚɤ
ɬɨɪɵ
f ɢ g ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɜɟɤɬɨɪɭ
ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɨɜ
Ɋ(0,0) – ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɬɨ, ɜ ɫɢɥɭ ɥɟɦɦɵ, ɜɟɤɬɨɪɵ
·
¸
ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵ ɢ ɢɦɟɸɬ ɪɚɜɧɵɟ ɞɥɢɧɵ.
¸
¸
¹
Ɋ(0,0) ɬɨɱɟɤ Ⱥ, ȼ, ɋ, ɬɨ x
+ x2 + x3 = y1 + y2 + y3 = 0, ɡɧɚɱɢɬ, ɜɟɤ-
1
1
·
§
¸
¨
(fe = 0 ɢ ɬɚɤ ɤɚɤ ge = 0), ɬɨ ɟɫɬɶ
11
e
¸
¨
¸
¨
¹
©
f ɢ g ɥɟɠɚɬ ɜ ɩɥɨɫɤɨɫɬɢ x + y + z = 0. ɇɚɣɞɟɦ ɜɫɟ ɬɚɤɢɟ
, ɭ1), ȼ(ɯ2, ɭ2)
1
f
ɜɟɤɬɨɪɵ.
ȼɵɛɟɪɟɦ ɩɚɪɭ ɩɪɨɢɡɜɨɥɶɧɵɯ ɨɪɬɨɝɨɧɚɥɶɧɵɯ ɜɟɤɬɨɪɨɜ
ɞɥɢɧɚɦɢ ɢ ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɦɢ ɜɟɤɬɨɪɭ
ɟ, ɧɚɩɪɢɦɟɪ,
e
ɟ
, ɟ2 ɫ ɪɚɜɧɵɦɢ
1
1
§
·
¨
¸
e
ɢ
1
¨
1
¨
2
©
2
¸
¸
¹
Ʌɟɝɤɨ ɩɪɨɜɟɪɢɬɶ, ɱɬɨ ɬɪɟɭɝɨɥɶɧɢɤ ɫ ɜɟɪɲɢɧɚɦɢ
ɹɜɥɹɟɬɫɹ ɪɚɜɧɨɫɬɨɪɨɧɧɢɦ ɫɨ ɫɬɨɪɨɧɨɣ
6
.
·
¸
¹
§
¨
¨
¨
¨
©
n
2
x
§
¨
x
¨
¨
x
©
03
·
1
¸
ɢ
¸
2
¸
3
¹
·
3
¸
¸
.
¸
¸
¹
)0 ,2(),3 ,1(),3 ,1( CBA
85
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ⱦɚɥɟɟ ɨɱɟɜɢɞɧɨ, ɱɬɨ ɥɸɛɚɹ ɞɪɭɝɚɹ ɩɚɪɚ f, g ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɵɯ ɜɟɤɬɨ-
6
ɪɨɜ, ɫ ɞɥɢɧɚɦɢ ɪɚɜɧɵɦɢ
ɟ
ɬɨɪɨɜ
+ɟ
, ɟ2 ɩɨɜɨɪɨɬɨɦ ɧɚ ɧɟɤɨɬɨɪɵɣ Į, ɬɨ ɟɫɬɶ f = ɟ1cosĮ – ɟ2sinĮ ɢ g = ɟ1sinĮ+
1
cosĮ. ɇɨ, ɬɨɝɞɚ ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɤɨɨɪɞɢɧɚɬɵ ɧɨɜɵɯ ɬɪɟɯ ɜɟɪɲɢɧ, ɩɨɥɭ-
2
ɱɟɧɧɵɯ ɢɡ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪɨɜ
ɜɟɪɲɢɧ
Ⱥ, ȼ, ɋ ɩɨɜɨɪɨɬɨɦ ɧɚ ɷɬɨɬ ɠɟ ɭɝɨɥ Į, ɬɨ ɟɫɬɶ ɨɛɪɚɡɭɸɬ ɪɚɜɧɨɫɬɨɪɨɧ-
ɢ ɨɪɬɨɝɨɧɚɥɶɧɵɦɢ ɜɟɤɬɨɪɭ ɟ, ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɜɟɤ-
f ɢ g, ɛɭɞɭɬ ɬɚɤɠɟ ɩɨɥɭɱɚɬɶɫɹ ɢɡ ɤɨɨɪɞɢɧɚɬ
ɧɢɣ ɬɪɟɭɝɨɥɶɧɢɤ.
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɬɟɨɪɟɦɚ 27 ɧɟ ɨɛɨɛɳɚɟɬɫɹ ɧɚ ɫɥɭɱɚɣ
§
·
§
ɪɚɫɫɦɨɬɪɟɜ ɜɟɤɬɨɪɵ
ɩɨɥɭɱɢɦ, ɱɬɨ ɬɨɱɤɚ
6
¨
¨
e
6
¨
1
0
¨
¨
0
©
Ɋ(0,0) ɛɭɞɟɬ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ ȼȽ7, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɫɥɟ-
¨
¸
¨
¸
e
,
¨
2
¸
¸
¸
¹
¨
¨
©
ɞɭɸɳɢɯ ɧɚɛɨɪɨɜ ɬɨɱɟɤ:
3 ,0,1 ,0,1 ,6,1 ,6
AAAA
. ɇɨ ɬɨɱɤɢ ɧɢ ɨɞɧɨɝɨ ɢɡ ɷɬɢɯ ɧɚɛɨɪɨɜ ɧɟ
4321
·
2
¸
¸
2
ɢ
¸
22
¸
¸
0
¹
n > 3. ȼ ɫɚɦɨɦ ɞɟɥɟ,
1
·
§
¸
¨
1
¸
¨
e
3
, ɜ ɫɢɥɭ ɥɟɦɦɵ,
¸
¨
1
¸
¨
¸
¨
3
¹
©
0 ,0,22 ,0,2 ,6,6 ,6
AAAA
4321
ɨɛɪɚɡɭɸɬ ɩɪɚɜɢɥɶɧɵɣ ɱɟɬɵɪɟɯɭɝɨɥɶɧɢɤ, ɯɨɬɹ ɢ ɢɧɬɟɪɟɫɧɵ.
ȼ ɡɚɤɥɸɱɟɧɢɟ ɧɟɥɶɡɹ ɧɟ ɭɩɨɦɹɧɭɬɶ ɡɚɞɚɱɢ, ɤɨɬɨɪɵɟ ɜɨɡɧɢɤɥɢ ɢɡ ɩɪɟɞɵɞɭɳɢɯ ɪɟɡɭɥɶɬɚɬɨɜ ɢ ɪɚɛɨɬɵ ɥɟɬɧɟɝɨ ɩɪɨɮɢɥɶɧɨɝɨ ɥɚɝɟɪɹ 2014 ɝ. ɜ ȽɈɍ Ʌɢɰɟɣ ʋ 1303. Ɍɟɨɪɟɦɵ, ɩɪɢɜɟɞɟɧɧɵɟ ɧɢɠɟ, ɛɵɥɢ ɞɨɤɚɡɚɧɵ ɭɱɟɧɢɤɨɦ 10-ɝɨ
ɤɥɚɫɫɚ ɂɜɚɧɨɦ Ɏɥɟɝɨɧɬɨɜɵɦ. ɇɚɡɨɜɟɦ ɷɬɢ ɡɚɞɚɱɢ
ɉɭɫɬɶ
Ⱥ
, …, Ⱥ
0
– ɜɟɪɲɢɧɵ ɩɪɚɜɢɥɶɧɨɝɨ ɦɧɨɝɨɭɝɨɥɶɧɢɤɚ, Ɇ – ɬɨɱɤɚ
n–1
ȼȽ9 ɢ ȼȽ10.
ɨɤɪɭɠɧɨɫɬɢ, ɨɩɢɫɚɧɧɨɣ ɨɤɨɥɨ ɷɬɨɝɨ ɦɧɨɝɨɭɝɨɥɶɧɢɤɚ, ɢ ɥɟɠɚɳɚɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɧɚ ɞɭɝɟ
Ⱥ0Ⱥ
; l – ɩɪɹɦɚɹ, ɩɪɨɯɨɞɹɳɚɹ ɱɟɪɟɡ ɰɟɧɬɪ ɷɬɨɣ ɨɤɪɭɠɧɨɫɬɢ.
1
r(A, l) – ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɬɨɱɤɢ Ⱥ ɞɨ ɩɪɹɦɨɣ l.
Ɂɚɞɚɱɚ ȼȽ9. ɉɪɢ ɤɚɤɢɯ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɥɚɯ n ɢ ɪ ɫɭɦɦɵ:
ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɜɵɛɨɪɚ ɩɪɹɦɨɣ
İ
= ±1, ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ ɥɟɠɢɬ ɥɢ ɬɨɱɤɚ Ⱥj ɧɚɞ ɢɥɢ ɩɨɞ ɩɪɹɦɨɣ l.
j
Ɂɚɞɚɱɚ ȼȽ10. ɉɪɢ ɤɚɤɢɯ ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɥɚɯ n ɢ ɪ ɫɭɦɦɵ:
1
n
¦
j
0
ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɜɵɛɨɪɚ ɩɪɹɦɨɣ
1
n
p
¦
j
0
),(),,(
lArlnps
ɢ
j
0
1
n
p
H
¦
j
0
),(),,(
lArlnps
j
j
l ɢ ɱɟɦɭ ɨɧɢ ɪɚɜɧɵ ɜ ɷɬɨɦ ɫɥɭɱɚɟ? Ɂɞɟɫɶ ɱɢɫɥɚ
1
p
||),,(
MAMnpsm
ɢ
j
n
¦
100
j
2
p
jpp
||)1(||||),,(
MAMAMAMnpsm
j
l ɢ ɱɟɦɭ ɨɧɢ ɪɚɜɧɵ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ?
Ɂɚɞɚɱɚ ȼȽ10 ɛɵɥɚ ɩɨɫɬɚɜɥɟɧɚ Ⱥ. Ƚ. Ɇɹɤɢɲɟɜɵɦ ɜɨ ɜɪɟɦɹ ɱɬɟɧɢɹ ɥɟɤɰɢɢ
ɪɚɛɨɬɵ ɥɟɬɧɟɝɨ ɩɪɨɮɢɥɶɧɨɝɨ ɥɚɝɟɪɹ 2014 ɝ. ɜ ȽɈɍ Ʌɢɰɟɣ ʋ 1303. ɂɦ ɛɵɥɢ
ɪɚɫɫɦɨɬɪɟɧɵ ɱɚɫɬɧɵɟ ɫɥɭɱɚɢ ɡɚɞɚɱɢ ɢ ɭɤɚɡɚɧɵ ɧɟɤɨɬɨɪɵɟ ɩɭɬɢ ɟɟ ɪɟɲɟɧɢɹ.
Ɂɞɟɫɶ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɬɨɥɶɤɨ ɱɚɫɬɧɵɟ ɫɥɭɱɚɢ ɷɬɢɯ ɡɚɞɚɱ, ɚ ɢɦɟɧɧɨ ɫɭɦɦɵ s(p, n, l) ɢ sm(p, n, M) ɪɚɫɫɦɨɬɪɢɦ ɞɥɹ ɱɟɬɧɵɯ ɪ (p = 2k), ɚ ɜɵɪɚɠɟɧɢɹ
s
(p, n, l) ɢ sm0(p, n, M) – ɞɥɹ ɧɟɱɟɬɧɵɯ. ɋɮɨɪɦɭɥɢɪɭɟɦ ɢ ɞɨɤɚɠɟɦ ɫɥɟɞɭɸɳɢɟ
0
ɬɟɨɪɟɦɵ.
ɢ
86
ɉɈɇəɌɂȿ Ɉ ȼɕɑɂɋɅɂɌȿɅɖɇɈɃ ȽȿɈɆȿɌɊɂɂ
Ɍɟɨɪɟɦɚ 28 (ɡɚɞɚɱɚ ȼȽ9). ɉɭɫɬɶ n > 1 ɢ p = 2k – ɧɚɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ. Ɍɨ-
ɝɞɚ
1
n
p
¦
0
j
) ,() , ,(
lArlnps
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɩɪɹɦɨɣ ɬɨɥɶɤɨ ɩɪɢ ɜɫɟɯ ɱɟɬɧɵɯ
j
n>2k ɢɥɢ ɧɟɱɟɬɧɵɯ n > k. ɉɪɢ ɷɬɨɦ
k
2
§
·
n
¨
{
lnks
),,2(
¸
k
2
¨
¸
k
2
©
¹
.
Ɍɟɨɪɟɦɚ 29 (ɡɚɞɚɱɚ ȼȽ9). ɉɭɫɬɶ p = 2k + 1 (k 0) – ɧɟɱɟɬɧɨɟ ɱɢɫɥɨ. Ɍɨ-
ɝɞɚ ɫɭɦɦɵ
0
¦
1
n
p
H
j
0
j
),(),,(
lArlnps
= 0, ɩɪɢ ɜɫɟɯ ɱɟɬɧɵɯ
j
n ɢ ɜɫɟɯ ɧɟɱɟɬɧɵɯ
n > p ɢ ɩɪɹɦɵɯ l, ɩɪɨɯɨɞɹɳɢɯ ɱɟɪɟɡ ɰɟɧɬɪ ɨɩɢɫɚɧɧɨɣ ɨɤɪɭɠɧɨɫɬɢ. Ⱦɥɹ ɞɪɭ-
ɝɢɯ
n ɷɬɢ ɫɭɦɦɵ ɡɚɜɢɫɹɬ ɨɬ ɩɪɹɦɨɣ l (ɱɢɫɥɚ İ
ɠɢɬ ɥɢ ɬɨɱɤɚ
Ɍɟɨɪɟɦɚ 30 (ɡɚɞɚɱɚ ȼȽ10). ɉɭɫɬɶ n > 1 ɢ p = 2k – ɧɚɬɭɪɚɥɶɧɵɟ ɱɢɫɥɚ.
Ɍɨɝɞɚ
Ⱥ
ɧɚɞ ɢɥɢ ɩɨɞ ɩɪɹɦɨɣ l, ɪɚɜɧɵ ɧɭɥɸ).
j
1
n
¦
0
j
p
||),,(
MAMnpsm
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɵɛɨɪɚ ɬɨɱɤɢ M, ɬɨɥɶɤɨ ɟɫɥɢ
j
= ±1, ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɥɟ-
j
n > k. ɉɪɢ ɷɬɨɦ
k
2
§
·
¨
¸
{
nlnks
),,2(
.
¨
¸
k
©
¹
Ɍɟɨɪɟɦɚ 31 (ɡɚɞɚɱɚ ȼȽ10). ɉɭɫɬɶ p = 2k + 1 (k 0) – ɧɟɱɟɬɧɨɟ ɱɢɫɥɨ. Ɍɨ-
ɝɞɚ ɫɭɦɦɵ
n ɛɨɥɶɲɢɯ ɪ = 2k + 1 ɢ ɬɨɱɟɤ Ɇ, ɥɟɠɚɳɢɯ ɧɚ ɞɭɝɟ Ⱥ0Ⱥ
ɧɵɯ
ɫɭɦɦɵ ɡɚɜɢɫɹɬ ɨɬ ɬɨɱɤɢ
Ɇ.
1
n
¦
100
2
j
p
jpp
||)1(||||),,(
MAMAMAMnpsm
= 0, ɞɥɹ ɥɸɛɵɯ ɧɟɱɟɬ-
j
. Ⱦɥɹ ɞɪɭɝɢɯ n ɷɬɢ
1
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɷɬɢɯ ɬɟɨɪɟɦ ɜɜɟɞɟɦ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɬɚɤ ɱɬɨ
ɜɟɪɲɢɧɵ ɩɪɚɜɢɥɶɧɨɝɨ
n-ɭɝɨɥɶɧɢɤɚ ɢɦɟɸɬ ɜɢɞ:
SS
2
§
cos nnj
A
¨
j
©
2
j
n
j
·
sin,
¸
n
¹
...,,1...,0,
ɉɭɫɬɶ ɬɨɱɤɚ
ɜɢɞ:
xcosĮ + ysinĮ = 0. ɂɯ ɫɨɨɛɪɚɠɟɧɢɣ ɫɢɦɦɟɬɪɢɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ
ª
D
«
¬
Ɇ(cosĮ, sinĮ) ɢ ɩɪɹɦɚɹ l, ɩɪɨɯɨɞɹɳɚɹ ɱɟɪɟɡ ɬɨɱɤɭ (0,0), ɢɦɟɟɬ
S
2
º
,0
. Ʉɪɨɦɟ ɬɨɝɨ, ɧɚɦ ɩɨɧɚɞɨɛɹɬɫɹ ɧɟɤɨɬɨɪɵɟ ɮɨɪɦɭɥɵ, ɮɚɤɬɵ:
»
n
¼
87
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
k
2
1.
k
12
k
2
¦
j
2cos)(cos
j
1
j
k
¦
j
0
j
k
jk
j
1
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɷɬɢɯ ɮɨɪɦɭɥ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɮɨɪɦɭɥɚɦɢ
ee
x
ɗɣɥɟɪɚ:
cos
2
2
k
·
§
1
jtat
¸
¨
2
k
¸
¨
k
2
¹
©
j
2
§
1
2cos)1()(sin
jtat
¨
2
k
¨
2
©
x
sin,
!
..., ,1,0 ,
j
!
k
·
¸
¸
k
¹
ixixixix
ee
, ɝɞɟ i – ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ (i2 = –1),
2
kja
..., ,0 ,0,)12cos()(cos
kjbtjbt
k
12
jk
¦¦
j
0
j
.)12sin()1()(sin ,
tjbt
ɢ ɮɨɪɦɭɥɨɣ ɛɢɧɨɦɚ ɇɶɸɬɨɧɚ. ɇɚɩɪɢɦɟɪ, ɞɨɤɚɠɟɦ ɩɟɪɜɭɸ:
§
2
k
¨
x
cos
¨
©
k
1
2
k
2
kj
2
k
ixix
2
k
·
ee
¸
¸
2
¹
1
2
ijxjk
eC
2
k
2
1
j
2
k
2
k
2
0
j
k
2
§
§
·
¨
¨
¸
2
k
¨
¸
¨
k
©
¹
©
k
jk
¦¦
2
k
1
j
2
k
1
)2(
xjkiijx
¦¦
2
k
2
0
j
·
¸
.2cos2
jxC
¸
¹
)22(
xjkij
eCeeC
2
k
Ɉɫɬɚɥɶɧɵɟ ɮɨɪɦɭɥɵ ɥɟɝɤɨ ɜɵɜɟɫɬɢ, ɟɫɥɢ ɜɫɩɨɦɧɢɬɶ, ɱɬɨ
S
§
·
cossin S
¨
¸
2
©
¹
j
xjxxx
cos)1()cos(,
ɢ
)12cos()12cos(
xjxj
2coscos
jxx
2.
Ɍɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɟ ɫɢɫɬɟɦɵ {1, sinx, cosx, sin2x, …, sinnx, cosnx} ɥɢ-
.
2
ɧɟɣɧɨ ɧɟɡɚɜɢɫɢɦɵ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ
n
¦
0
j
1
jj
0
{
jj
....,,2,1,00)sincos(
njbaajxbjxaa
1
n
3.
l
0
cos
S
2
kl
§
¨
n
©
ɧɟ ɞɟɥɢɬɫɹ ɧɚ
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɷɬɨɝɨ ɭɬɜɟɪɠɞɟɧɢɹ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ ɗɣɥɟɪɚ:
n
¦
q
ix
inx
1
e
iqx
e
ix
e
0
sincos
1
1
1
n
kq
2
·
§
cos
0
q
¸
¨
n
¹
©
1
n
·
D
¸
¦¦
l
¹
S
2
kl
§
sin
¨
0
©
·
D
n
D
0
¸
¹
ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ k
n.
xixe
ɢ ɮɨɪɦɭɥɨɣ ɫɭɦɦɵ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ:
,...2,1,0,2,
mmx
S
1
n
i
sin
0
q
rr z
. Ɍɨɝɞɚ
q
2
kq
1
n
kq
SS
2
§
¨
n
©
i
·
¸
¹
n
0
q
2
k
SS
1
n
i
§
·
n
¨
¸
¦¦¦¦
¨
0
q
©
¸
¹
88
.ɧɚɞɟɥɢɬɫɹɧɟ0
nkee
ɉɈɇəɌɂȿ Ɉ ȼɕɑɂɋɅɂɌȿɅɖɇɈɃ ȽȿɈɆȿɌɊɂɂ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 28. Ɂɚɦɟɬɢɦ, ɱɬɨ
1
n
S
q
D
nps
2
cos),,(
n
0
q
S
2
D
sincos
n
p
1
q
n
D
¦¦
q
S
2
§
cossin
¨
n
0
©
p
q
·
(29)
D
¸
¹
Ɉɬɫɸɞɚ ɩɪɢ
ɪ = 2k ɢɡ ɮɨɪɦɭɥ 1 ɢɦɟɟɦ
n
k
1
1
§
¨
D
¦¦
¨
q
j
0
kjn
a
¦¦
110
©
4
qj
S
§
cos
¨
j
n
©
q
4
qj
S
§
cos),,2(
anks
j
0
2
j
D
2
j
D
¨
n
©
2
k
·
§
1
·
¸
¹
¸
¨
2
k
¸
¨
k
2
¹
©
2
k
·
·
§
1
·
¸
¹
.
n
¸
¸
¨
k
2
¸
¨
¸
k
2
¹
©
¹
ȼ ɫɢɥɭ 2. ɢ 3. ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɛɭɞɟɬ ɪɚɜɧɨ 0 ɩɪɢ ɜɫɟɯ Į, ɟɫɥɢ ɬɨɥɶɤɨ
ɜɫɟ ɱɟɬɧɵɟ ɱɢɫɥɚ 2, 4, …, 2
ɡɚɜɢɫɢɬ ɨɬ
Į ɬɨɥɶɤɨ ɩɪɢ ɜɫɟɯ ɱɟɬɧɵɯ n > 2k ɢ ɧɟɱɟɬɧɵɯ n > k. Ɍɟɨɪɟɦɚ 28
k, ɧɟ ɛɭɞɭɬ ɞɟɥɢɬɶɫɹ ɧɚ n. Ɍɨ ɟɫɬɶ s(2k, n, Į) ɧɟ
ɞɨɤɚɡɚɧɚ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 29. ɂɡ (29) ɢ ɮɨɪɦɭɥ 1. ɂɦɟɟɦ
0
kjn
b
¦¦
010
§
¨
D
¦¦
¨
00
©
S
§
cos
¨
j
q
n
©
§
cos),,12(
bnks
¨
j
©
j
)12(2
jq
)12(2
jq
S
n
·
.)12(
j
D
¸
¹
·
·
¸
)12(
j
D
¸
¸
¹
¹
nqk
1
ȼ ɫɢɥɭ 2. ɢ 3., ɟɫɥɢ ɯɨɬɹ ɛɵ ɨɞɧɨ ɢɡ ɱɢɫɟɥ 3, 5, …, 2
ɧɚ
n, ɬɨ ɷɬɨ ɜɵɪɚɠɟɧɢɟ (s
ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ. Ɍɨ ɟɫɬɶ
ɱɟɬɧɵɯ
n ɛɨɥɶɲɢɯ ɪ = 2k + 1 (k 0). Ɍɟɨɪɟɦɚ 29 ɞɨɤɚɡɚɧɚ.
) ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɨɬ Į ɢ ɛɭɞɟɬ ɪɚɜɧɨ ɧɭɥɸ
0
s
(2k + 1, n, Į) = 0 ɩɪɢ ɜɫɟɯ ɱɟɬɧɵɯ n ɢ ɜɫɟɯ ɧɟ-
0
k + 1 ɛɭɞɟɬ ɞɟɥɢɬɶɫɹ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 30. Ʉɚɤ ɢ ɜɵɲɟ, ɞɥɹ M(cos Į, sin Į) ɢɦɟɟɦ
n
D
j
1
n
1
§
¨
q
0
©
2
k
·
§
¸
¨
¸
¨
k
¹
©
2
q
S
§
cos22
¨
n
©
k
¦¦
j
1
n
1
§
2
q
S
§
p
MAnpsm
j
k
·
·
D
¸
¸
¹
¹
n
1
jk
an
j
q
0
§
¨
cos||),,(
¨
¨
¦¦
¨
n
©
q
0
©
©
n
1
§
kk
2
sin4
¨
©
q
0
2
jq
S
§
cos)1(4
¨
©
j
D
n
·
·
D
¸
¸
¹
¹
q
·
¸
2
n
¹
2
k
·
§
·
¸
¹
.
n
¸
¨
¸
¨
k
¹
©
§
sincos
¨
©
nqk
¦¦¦¦
2
§
¨
©
1
j
01
q
S
·
¸
n
¹
jk
a
sin
k
22
·
·
¸
D
¸
¸
¹
¹
2
jq
SDS
§
cos)1(4
j
¨
n
©
Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ, ɱɬɨ
ɥɚ 3, …,
k ɧɟ ɛɭɞɭɬ ɞɟɥɢɬɶɫɹ ɧɚ n (ɬɨɝɞɚ ɜ ɫɢɥɭ 3. ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ
ɜ ɩɨɫɥɟɞɧɟɦ ɪɚɜɟɧɫɬɜɟ ɛɭɞɟɬ ɪɚɜɧɨ ɧɭɥɸ). Ɍɨ ɟɫɬɶ ɬɨ
ɨɬ
Į, ɬɨɥɶɤɨ ɟɫɥɢ n > k ɜ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɡɚɜɢɫɢɬ. Ɍɟɨɪɟɦɚ 30 ɞɨɤɚɡɚɧɚ.
sm(2k, n, Į) ɧɟ ɡɚɜɢɫɢɬ ɨɬ Į, ɟɫɥɢ ɬɨɥɶɤɨ ɜɫɟ ɱɢɫ-
s(2k, n, Į) ɧɟ ɡɚɜɢɫɢɬ
·
j
D
¸
¹
89
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 31. Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɩɨɥɨɠɢɦ q = 1, 2, …, n
S
dDd20
ɢ
q
§
¨
©
, ɬɨɝɞɚ
n
§
2
q
S
§
12
p
cos22
k
12
¦
j
0
k
12
¦
j
0
k
¨
MAMA
q
2
q
S
§
¨
n
©
jk
b
jk
b
cos||||
¨
¨
©
©
p
2
·
·
D
¸
¸
¹
¹
§
sin)1(2
¨
j
©
§
sin)1(2
¨
j
©
·
§
¸
¨
n
¹
©
sin2
jq
n
)12(
jq
cos
n
·
D
¸
¹
q
§
kk
1212
¨
n
©
DS
)12()12(
j
2
)12(
2
2
q
S
§
sincos
¨
©
DS
·
¸
2
¹
·
¸
¹
cos
·
§
¨
©
sin
¸
n
¹
jqj
n
p
22
2
·
·
¸
D
¸
¸
¹
¹
)12(
sin
)12(
j
DSDS
·
¸
2
¹
Ⱦɚɥɟɟ ɪɚɫɫɦɨɬɪɢɦ ɫɭɦɦɵ:
1
n
q
2
q
1
n
q
¦¦
2
q
S
xqxnxxqxnxx
ɝɞɟ,cos)1(coscosɢsin)1(sinsin
ɉɪɢɦɟɧɹɹ ɮɨɪɦɭɥɭ ɗɣɥɟɪɚ, ɩɨɥɭɱɚɟɦ
ix
e
1
n
l
1
ilxlinxix
2
)1(
ixinxn
ee
ix
1
S
ɝɞɟ,)1(
xeee
ix
ee
)12(
j
n
12
n
1)1(
1
n
1
lixix
)(1
ee
¦¦
2
l
z
xn
Ɉɬɫɸɞɚ ɢ (30) ɜɵɬɟɤɚɟɬ, ɱɬɨ
1
12
00
1
n
¦
q
qkk
2
1212
k
MAMAMAMnksm
q
ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ
n ɧɟɱɟɬɧɨɟ ɢ ɛɨɥɶɲɟ ɪ = 2k + 1.
Ɍɟɨɪɟɦɚ 31 ɞɨɤɚɡɚɧɚ.
(30)
.
j
n
SS
0||)1(||||),,12(
)12(
.
.2modɢɧɟɱɟɬɧɨɟɟɫɥɢ,0
90
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