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ǾȇȘșȣ 3. ǲǯǴǬǰǴǧȆ ǧǲǪǬǨǷǧ

3.1. ǱȕȓȖȒȌȑȘȔȢȌ ȞȏȘȒȇ

Ɉɩɪɟɞɟɥɟɧɢɟ Ʉɨɦɩɥɟɤɫɧɵɦɢ ɱɢɫɥɚɦɢ ɧɚɡɵɜɚɟɬɫɹ ɦɧɨɠɟɫɬɜɨ ɭɩɨ ɪɹɞɨɱɟɧɧɵɯ ɩɚɪ ɜɟɳɟɫɬɜɟɧɧɵɯ ɱɢɫɟɥ (x, y) ɞɥɹ ɤɨɬɨɪɵɯ ɜɜɟɞɟɧɵ ɨɩɟɪɚɰɢɢ ɫɥɨɠɟɧɢɹ ɢ ɭɦɧɨɠɟɧɢɹ ɩɨ ɫɥɟɞɭɸɳɢɦ ɩɪɚɜɢɥɚɦ:

1.(x1, y1) (x2, y2) (x1 x2, y1 y2) ,

2.(x1, y1)¸(x2, y2) (x1 ¸x2 y1 ¸ y2, x1 ¸ y2 x2 ¸ y1) .

Ʉɨɦɩɥɟɤɫɧɵɟ ɱɢɫɥɚ z1 (x1, y1) ɢ z2 (x2, y2) ɧɚɡɵɜɚɸɬɫɹ ɪɚɜɧɵɦɢ ɬɨ ɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ ɤɨɝɞɚ

x1 x2, y1 y2 .

ǸȉȕȐȘșȉȇ ȕȖȌȗȇȝȏȐ Ȕȇȋ ȑȕȓȖȒȌȑȘȔȢȓȏ ȞȏȘȒȇȓȏ

1.z1 z2 z2 z1.

2.(z1 z2) z3 z1 (z2 z3) .

3.z1 ¸z2 z2 ¸z1 .

4.(z1 ¸z2)¸z3 z1 ¸(z2 ¸z3) .

5.z1 ¸(z2 z3) z1 ¸z2 z1 ¸z3 .

Ɉɩɟɪɚɰɢɹ ɜɵɱɢɬɚɧɢɹ ɞɥɹ ɱɢɫɟɥ z1 (x1, y1) ɢ z2 (x2, y2) ɨɩɪɟɞɟɥɹɟɬɫɹ

ɤɚɤ

z1 z2 (x1, y1) ( x2, y2) (x1 x2, y1 y2) .

Ɉɩɟɪɚɰɢɹ ɞɟɥɟɧɢɹ ɩɪɢ x 2			y	2 v0):
		2		2
	z	x ¸x	y ¸ y				y ¸x x ¸ y			¬
	z	x ¸x	y ¸ y		2		y ¸x x ¸ y		2
	1	1 2		1	2	,	1 2	1	2		.
		2		2		,	2	2			.
		2		2			2	2
		x2	y2				x2 y2
	z2	x2	y2				x2 y2			®

ɍɩɨɪɹɞɨɱɟɧɧɵɟ ɩɚɪɵ (x, y) ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɟɤ ɩɥɨɫɤɨɫɬɢ Ɍ ɟ ɤɚɠɞɨɣ ɬɨɱɤɟ ɩɥɨɫɤɨɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɟɤɨɬɨɪɨɟ ɤɨɦɩɥɟɤɫ ɧɨɟ ɱɢɫɥɨɢ ɨɛɪɚɬɧɨ ɤɚɠɞɨɦɭɤɨɦɩɥɟɤɫɧɨɦɭɱɢɫɥɭɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɨɱɤɚ ɩɥɨɫ ɤɨɫɬɢ ɉɥɨɫɤɨɫɬɶ ɧɚ ɤɨɬɨɪɨɣ ɢɡɨɛɪɚɠɟɧɵ ɤɨɦɩɥɟɤɫɧɵɟ ɱɢɫɥɚ ɧɚɡɵɜɚɟɬɫɹ ɤɨɦɩɥɟɤɫɧɨɣ ɩɥɨɫɤɨɫɬɶɸ

Ɂɚɦɟɬɢɦ ɱɬɨ ɫɤɥɚɞɵɜɚɹ ɢ ɭɦɧɨɠɚɹ ɤɨɦɩɥɟɤɫɧɵɟ ɱɢɫɥɚ ɜɢɞɚ z1 (x1,0) ɢ z2 (x2,0) ɦɵ ɩɨɥɭɱɚɟɦ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ ɬɚɤɨɝɨ ɠɟ ɜɢɞɚ

z1 z2 (x1 x2,0) , z1 ¸z2 (x1 ¸x2,0) .

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɦɵ ɦɨɠɟɦ ɨɬɨɠɞɟɫɬɜɢɬɶ ɬɨɱɤɢ ɥɟɠɚɳɢɟ ɧɚ ɤɨɨɪɞɢɧɚɬ ɧɨɣ ɨɫɢ Ox, ɫ ɜɟɳɟɫɬɜɟɧɧɵɦɢ ɱɢɫɥɚɦɢ (x,0) x ɇɚɩɪɢɦɟɪ, (1,0) 1. ȼɨɡɜɟ

ɞɟɦ ɜ ɤɜɚɞɪɚɬ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ (0,1) :

(0,1)2 (0,1)¸(0,1) (0¸0 1¸1,0¸1 1¸0) ( 1,0) 1.

ȼɢɞɢɦ ɱɬɨɧɚ ɦɧɨɠɟɫɬɜɟɤɨɦɩɥɟɤɫɧɵɯ ɱɢɫɟɥ ɫɭɳɟɫɬɜɭɟɬ ɤɨɪɟɧɶ ɢɡ ɜɟɳɟ ɫɬɜɟɧɧɨɝɨ ɱɢɫɥɚ 1 ɗɬɨ ɱɢɫɥɨ ɧɚɡɵɜɚɟɬɫɹ ɦɧɢɦɨɣ ɟɞɢɧɢɰɟɣ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ i (0,1) .

Ʌɸɛɨɟ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ

z(x, y) x¸(1,0) y ¸(0,1) x iy .

xɧɚɡɵɜɚɟɬɫɹ ɜɟɳɟɫɬɜɟɧɧɨɣ ɱɚɫɬɶɸ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

xRe z, y ɧɚɡɵɜɚɟɬɫɹ ɦɧɢɦɨɣ ɱɚɫɬɶɸ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ

yIm z Ɂɚɩɢɫɶ

z x iy

ɧɚɡɵɜɚɟɬɫɹ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɡɚɩɢɫɶɸ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ

ǷȏȘ

ɑɢɫɥɨ z x iy ɧɚɡɵɜɚɟɬɫɹ ɫɨɩɪɹɠɟɧɧɵɦ ɤ ɱɢɫɥɭ z x iy. ɉɪɨɢɡɜɟɞɟ ɧɢɟ ɱɢɫɥɚ ɧɚ ɫɨɩɪɹɠɟɧɧɨɟ ɤ ɧɟɦɭ ɜɫɟɝɞɚ ɹɜɥɹɟɬɫɹ ɜɟɳɟɫɬɜɟɧɧɵɦ ɱɢɫɥɨɦ z ¸z (x iy)(x iy) x2 y2. ȼɟɪɧɟɦɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɪɚɧɟɟɨɩɟɪɚɰɢɢ ɞɟ ɥɟɧɢɹ ɤɨɦɩɥɟɤɫɧɵɯ ɱɢɫɟɥ Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɧɚɣɬɢ ɱɚɫɬɧɨɟ ɨɬ ɞɟɥɟɧɢɹ ɱɢɫɟɥ

z1 x1 iy1 ɢ z2 x2 iy2 ɧɭɠɧɨ ɱɢɫɥɢɬɟɥɶ ɢ ɡɧɚɦɟɧɚɬɟɥɶ ɞɪɨɛɢ ɭɦɧɨɠɢɬɶ ɧɚ ɱɢɫɥɨ ɫɨɩɪɹɠɟɧɧɨɟ ɤ ɡɧɚɦɟɧɚɬɟɥɸ

	z1	x1 iy1	(x1 iy1)(x2 iy2)		x1x2 y1y2 i(x2 y1 x1y2)		.
	z2		(x2 iy2)(x2 iy2)				.
	z2	x2 iy2	(x2 iy2)(x2 iy2)		x 2	y 2
				2		2
ɉɪɢɦɟɪ
			(2 3i) ( 1 4i) 1 7i

(1 2i)¸(3 5i) 1¸3 1¸5i 2i ¸3 2i ¸5i 3 5i 6i 10 7 11i

(1 i)2 12 2¸1¸i i2 1 2i 1 2i

2 3i

(2 3i)(3 i)

3 11i

3 i

(3 i)(3 i)

1 4i

( 1 4i)(2 i)

6 9i

2 i

(2 i)(2 i)

Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ȼɵɱɢɫɥɢɬɶ z

¸z

z 2

/ z

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

5 2i

3 i

1 3i

4 2i

7 2i

1 3i

6 5i

2 3i

4 2i

7 4i

3 3i

2 5i

4 7i

1 2i

5 3i

3 2i

1 10i

2 i

3 i

4 2i

3 7i

4 i

1 i

1 3i

4 5i

2 3i

3 7i

3 5i

4 2i

5 2i

4 i

1 3i

3 7i

1 3i

3 2i

1 3i

1 10i

12 4i

2 5i

7 4i

4 2i

7 2i

3 2i

1 i

3 i

4 5i

6 2i

3 4i

3 7i

1 i

4 i

3 5i

4 i

9 2i

2 5i

3 4i

ɉɭɫɬɶ ɧɚ ɤɨɦɩɥɟɤɫɧɨɣ ɩɥɨɫɤɨɫɬɢ ɡɚɞɚɧɚ ɩɨɥɹɪɧɚɹ ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ ɫɨɜɦɟɳɟɧɧɚɹ ɫ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɨɣ ɤɨɨɪɞɢɧɚɬ ɩ Ɍɨɝɞɚ ɬɨɱɤɚ z ɨɛɥɚ ɞɚɟɬ ɨɞɧɨɜɪɟɦɟɧɧɨ ɢ ɞɟɤɚɪɬɨɜɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ (x, y) ɢ ɩɨɥɹɪɧɵɦɢ (r,K)

ɉɨɥɹɪɧɵɣ ɪɚɞɢɭɫ r ɧɚɡɵɜɚɟɬɫɹ ɦɨɞɭɥɟɦ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɢ ɨɛɨɡɧɚɱɚɟɬ ɫɹ r z ɩɨɥɹɪɧɵɣ ɭɝɨɥ K ɧɚɡɵɜɚɟɬɫɹ ɚɪɝɭɦɟɧɬɨɦ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ Argz K Ⱥɪɝɭɦɟɧɬ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ 2Q ȿɫɥɢ ɱɢɫɥɨ K ɹɜɥɹɟɬɫɹ ɚɪɝɭɦɟɧɬɨɦ ɬɨ ɢ ɥɸɛɨɟ ɱɢɫɥɨ ɜɢɞɚ K 2Qk,k Z ɬɚɤɠɟ ɹɜɥɹɟɬɫɹ ɚɪɝɭɦɟɧɬɨɦ ɷɬɨɝɨ ɱɢɫɥɚ Ƚɥɚɜɧɵɦ ɡɧɚɱɟɧɢɟɦ ɚɪɝɭɦɟɧɬɚ argz ɧɚɡɵɜɚɟɬɫɹ ɧɚɢɦɟɧɶɲɢɣ ɩɨ ɚɛɫɨɥɸɬɧɨɣ ɜɟɥɢɱɢɧɟ ɚɪɝɭɦɟɧɬ argz (Q,Q] Ɂɚ ɩɢɫɶ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɜ ɜɢɞɟ

z z (cosK isin K)

ɧɚɡɵɜɚɟɬɫɹ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɡɚɩɢɫɶɸ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ Ɇɨɞɭɥɢ ɤɨɦ ɩɥɟɤɫɧɨ ɫɨɩɪɹɠɟɧɧɵɯ ɱɢɫɟɥ ɪɚɜɧɵ z z .

ȼ ɩ ɛɵɥɢ ɜɵɜɟɞɟɧɵ ɮɨɪɦɭɥɵ ɫɜɹɡɢ ɦɟɠɞɭ ɩɨɥɹɪɧɵɦɢ ɢ ɞɟɤɚɪɬɨ ɜɵɦɢ ɤɨɨɪɞɢɧɚɬɚɦɢ ɬɨɱɤɢ ɩɥɨɫɤɨɫɬɢ Ɍɟɩɟɪɶ ɦɵ ɦɨɠɟɦ ɢɦɢ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɥɹ ɫɜɹɡɢ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɢ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɵ ɡɚɩɢɫɢ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɉɭɫɬɶ ɤɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ ɡɚɩɢɫɚɧɨ ɜ ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɮɨɪɦɟ

z x iy.

Ɍɨɝɞɚ ɧɚɯɨɞɢɦ

z x2 y2 ,

K ɧɚɯɨɞɢɦ ɤɚɤ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ

£		2		2
¦	x	2	y	2
¦cosK x /	x		y
¦					.
¤					.
¦	x2 y2
¦sin K y /	x2 y2
¦
¥

ȿɫɥɢɤɨɦɩɥɟɤɫɧɨɟɱɢɫɥɨɡɚɞɚɧɨɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣɮɨɪɦɟ ɬɨɧɚɯɨɞɢɦ

x Re z z cosK, y Im z z sin K.

ɉɪɢɦɟɪ

z 2(cos(5Q / 4) isin(5Q/ 4)) 2( 2 / 2 i( 2 / 2)) 2 i 2.

Ɂɚɩɢɲɟɦ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ ɱɢɫɥɨ z i . Re z 0 Im z 1. Ɉɬɫɸɞɚ i 02 12 1. Ⱥɪɝɭɦɟɧɬ ɜɵɱɢɫɥɹɟɦ ɤɚɤ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ

¦cosK 0		Q
£
¦	K		.
¤
¤		2
¦sin K1		2
¥¦

ɉɨɥɭɱɚɟɦ

i cos(Q/ 2) isin(Q/ 2).

Ɂɚɩɢɲɟɦ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ ɱɢɫɥɨ z 2 2i.

Re z 2, Im z 2.

Ɉɬɫɸɞɚ z 22 ( 2)2 2 2 . Ⱥɪɝɭɦɟɧɬ ɜɵɱɢɫɥɹɟɦ ɤɚɤ ɪɟɲɟɧɢɟ ɫɢɫɬɟɦɵ

£		Q
¦cosK 1 / 2		Q
¦	K		.
¤
¤
¦	4
¦sin K1 / 2	4
¥
ɉɨɥɭɱɚɟɦ

2 2i 2 2(cos(Q/ 4) isin(Q/ 4).

Ɍɟɤɫɬ ɡɚɞɚɧɢɹ Ɂɚɩɢɫɚɬɶ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ ɱɢɫɥɨ

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

1	3 3 3i			2	2 2i			3	4 4		i
										3
4	2 2 3i			5	7		7i	6	2 3 2i
						3
7	2 2 3i			8	2 2i			9	2 3 2i

10		5i		11	2 2i			12	3 3 3i

13	4 4 3i			14	3 3i			15	5 3 5i
16	2 2 3i			17	3			18	4 4i
19		2i		20	4 4 3i			21	2 3 2i
22	2 3 2i			23	5 5 3i			24	4i

25			3i	26	2 2i			27	9 3 9i
		3
28		7 7i		29	8 8i			30	2 2 3i

Ɋɚɫɫɦɨɬɪɢɦɩɪɨɢɡɜɟɞɟɧɢɟɤɨɦɩɥɟɤɫɧɵɯɱɢɫɟɥ ɡɚɞɚɧɧɵɯɜɬɪɢɝɨɧɨɦɟɬɪɢ ɱɟɫɤɨɣ ɮɨɪɦɟ

z1 ¸z2 z1 (cosK1 isin K1) z2 (cosK2 isin K2)

z1 z2 (cosK1 cosK2 sin K1 sin K2 i(sin K1 cosK2 sin K2 cosK1))

z1 z2 (cos(K1 K2) isin(K1 K2)) .

Ɇɵ ɩɨɥɭɱɢɥɢ ɱɬɨ ɩɪɢ ɭɦɧɨɠɟɧɢɢ ɤɨɦɩɥɟɤɫɧɵɯ ɱɢɫɟɥ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢ ɱɟɫɤɨɣ ɮɨɪɦɟ ɢɯ ɦɨɞɭɥɢ ɭɦɧɨɠɚɸɬɫɹ ɚ ɚɪɝɭɦɟɧɬɵ ɫɤɥɚɞɵɜɚɸɬɫɹ ɉɨɥɶɡɭɹɫɶ ɷɬɢɦ ɮɚɤɬɨɦ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ

Ɏɨɪɦɭɥɭ Ɇɭɚɜɪɚ:

zn z (cosK isin K) n z n (cosnK isin nK).

ɉɪɢɦɟɪ ȼɵɱɢɫɥɢɦ 1 i 20 . Ɂɚɩɢɲɟɦ ɱɢɫɥɨ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ:

1 i 2(cos Q4 isin Q4).

ɉɨ ɮɨɪɦɭɥɟ Ɇɭɚɜɪɚ ɩɨɥɭɱɚɟɦ

1 i 20	( 2)20(cos	Q	¸20 isin	Q	¸20) 210(cos5Qisin5Q) 1024.
		4		4

ȼɵɱɢɫɥɢɦ 1 i 3 46 . Ɂɚɩɢɲɟɦ ɱɢɫɥɨ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ

1 i 3 2(cos( Q3) isin( Q3)).

ɉɨ ɮɨɪɦɭɥɟ Ɇɭɚɜɪɚ ɩɨɥɭɱɚɟɦ

1 i	3 46		246(cos(		Q	)¸46 isin(					Q	)¸46)

				3					3
246(cos	2Q	isin		2Q) 246(			1	i	3	) 245			( 1 i 3).
	3						2		2
				3

Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ȼɵɱɢɫɥɢɬɶ ɭɤɚɡɚɧɧɭɸ ɫɬɟɩɟɧɶ ɱɢɫɥɚ.

ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

1	(1 i)27	2	( 1 i)45
3	( 1 i)25	4	(1				i)34
						3
5	(1 3i)27	6	(3 3i)34
7	( 2 2i)23	8	( 3 / 2 i / 2)16
9	( 1/ 3 i / 3)32	10	(2 2i)40
11	( 2 2i)20	12	( 2		2i)12
				3
13	( 3 i)43	14	(1 3i)51
15	( 3 3 3i)14	16	(4 4i)13
17	( 1 i)43	18	( 3 i)25
19	(3 3i)35	20	( 1/ 2 i / 2)24
21	( 1 3i)23	22	(2 3 2i)10

23	(2 2	3i)23	24	(1 i)34
25	(4 4	3i)15	26	( 5 5i)32
27	( 3 i)33		28	(3 3i)37
29	(3 3i)47		30	( 3 i)43

Ɏɨɪɦɭɥɭ Ɇɭɚɜɪɚ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɤɨɪɧɟɣ ɫɬɟɩɟɧɢ n

ɢɡ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɉɭɫɬɶ w n z ɬɨɝɞɚ z wn, ɡɚɩɢɲɟɦ ɜ ɬɪɢɝɨɧɨɦɟɬ ɪɢɱɟɫɤɨɣ ɮɨɪɦɟ z r(cosK isin K), w S(cosR isin R) ɉɨ ɮɨɪɦɭɥɟ Ɇɭɚɜɪɚ ɩɨɥɭɱɚɟɦ

r(cosK isin K) Sn (cos nR isin nR).

Ɉɬɫɸɞɚ ɩɨɥɭɱɚɟɦ Sn r, nR K ɫɥɟɞɨɜɚɬɟɥɶɧɨ S n r R K / n Ɇɵ ɞɨɥɠɧɵ ɜɫɩɨɦɧɢɬɶ ɱɬɨ ɚɪɝɭɦɟɧɬ K ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɨɞ ɧɨɡɧɚɱɧɨ ɟɫɥɢ K ɹɜɥɹɟɬɫɹ ɚɪɝɭɦɟɧɬɨɦ ɱɢɫɥɚ z, ɬɨ K 2Qk,k Z ɬɚɤɠɟ ɹɜɥɹ ɟɬɫɹ ɚɪɝɭɦɟɧɬɨɦ ɱɢɫɥɚ z. Ɍ ɨ ɞɥɹ ɤɚɠɞɨɝɨ ɤɨɦɩɥɟɤɫɧɨɝɨ ɱɢɫɥɚ ɦɵ ɦɨɠɟɦ ɩɨɥɭɱɢɬɶ n ɪɚɡɥɢɱɧɵɯ ɤɨɪɧɟɣ ɫɬɟɩɟɧɢ n ɜɢɞɚ

n r(cosK isin K)			n r (cos			K 2Qk					isin			K 2Qk				),
n r(cosK isin K)			n r (cos					n			isin							),
								n								n
			k 0,1, ..., n 1.
ɉɪɢɦɟɪ ɇɚɣɞɟɦ				i Ɂɚɩɢɫɵɜɚɟɦ ɱɢɫɥɨ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ
ɮɨɪɦɟ
	i cos(Q/ 2) isin(Q/ 2).
Ʉɜɚɞɪɚɬɧɵɟ ɤɨɪɧɢ ɢɦɟɸɬ ɜɢɞ
i cos(		Q/ 2 2Qk			) isin(				Q/ 2 2Qk								)
i cos(					) isin(												)
				2							2		.
ɉɨɞɫɬɚɜɥɹɹ k 0,1 ɩɨɥɭɱɚɟɦ ɞɜɚ ɤɨɪɧɹ
w	cos(		Q	) isin(			Q	)			2	i	2		,
w	cos(			) isin(				)				i			,
1			4	4							2		2
			4	4							2		2
w	cos(3Q) isin(3Q)									2		i	2			.
w	cos(3Q) isin(3Q)											i				.
2	4			4							2		2
	4			4							2		2

ɇɚɣɞɟɦ 4 1 Ɂɚɩɢɫɵɜɚɟɦ ɱɢɫɥɨ ɜ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ

1 cosQisin Q.

Ʉɨɪɧɢ ɱɟɬɜɟɪɬɨɣ ɫɬɟɩɟɧɢ ɢɦɟɸɬ ɜɢɞ

		4 1 cos(						Q2Qk					) isin(						Q2Qk									).
		4 1 cos(											) isin(															).
									4											4
ɉɨɞɫɬɚɜɥɹɹ k 0,1, 2, 3, ɩɨɥɭɱɚɟɦ ɱɟɬɵɪɟ ɡɧɚɱɟɧɢɹ ɤɨɪɧɹ
		w cos					Q		isin			Q			2			i				2			,
		w cos							isin									i							,
			1			4			4						2					2
						4			4						2					2
		w		cos 3Q isin 3Q												2					i			2						,
		w		cos 3Q isin 3Q																	i									,
		2				4			4						2									2
						4			4						2									2
		w		cos 5Q isin 5Q													2				i			2					,
		w		cos 5Q isin 5Q																	i								,
		3				4			4						2									2
						4			4						2									2
		w cos 7Q							isin 7Q						2					i				2			.
		w cos 7Q							isin 7Q											i							.
			4			4			4						2								2
						4			4						2								2
ɇɚɣɞɟɦ n 1 ȼ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɨɣ ɮɨɪɦɟ 1 cos0 isin 0 .
Ʉɨɪɧɢ ɫɬɟɩɟɧɢ n ɢɦɟɸɬ ɜɢɞ
					n 1 cos(					2Qk	) isin(2Qk ).
					n 1 cos(						) isin(2Qk ).
										n									n
ɉɨɞɫɬɚɜɥɹɹ k 0,1, .., n 1 ɩɨɥɭɱɚɟɦ n ɪɚɡɥɢɱɧɵɯ ɤɨɪɧɟɣ ɜɫɟ ɤɨɪɧɢ ɥɟ
ɠɚɬ ɧɚ ɟɞɢɧɢɱɧɨɣ ɨɤɪɭɠɧɨɫɬɢ ɢ ɞɟɥɹɬ ɟɟ ɧɚ n ɪɚɜɧɵɯ ɞɭɝ ɞɥɢɧɵ																															2Q	.
																															n	.
																															n
Ɍɟɤɫɬ ɡɚɞɚɧɢɹ ȼɵɱɢɫɥɢɬɶ ɤɨɪɧɢ ɭɤɚɡɚɧɧɨɣ ɫɬɟɩɟɧɢ
ȼɚɪɢɚɧɬɵ ɡɚɞɚɧɢɹ ʋ

	1	6 64				2			4 81												3									3 i
	4	1 3i				5			12 1000												6					4 16
	7	9				8			3 8												9									(1/ i)
	10	6 9				11			3 8												12									4 4 3i
	13	2 2			i	14								i							15									5 5 3i
	13	2 2		3	i	14								i							15									5 5 3i
	16	12 27				17					1 3i										18									7 7 3i
	19	6 8				20				5 5 3i											21									3 (1/ i)
	22	3 i				23				2 2 3i											24					3 8
	25	(1/ i)				26				4 4 3i											27					16

	28	3 3 3i				29			3 27												30									3 (1/ i)

Ɋɚɫɫɦɨɬɪɢɦ ɚɥɝɟɛɪɚɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɫɬɟɩɟɧɢ n

anzn an 1zn 1 ... a1z a0 0. (*)

Ɂɞɟɫɶ a j , j 0,...,n – ɩɨɫɬɨɹɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ Ʉɨɦɩɥɟɤɫɧɨɟ ɱɢɫɥɨ

ɧɚɡɵɜɚɟɬɫɹ ɤɨɪɧɟɦ ɭɪɚɜɧɟɧɢɹ ɟɫɥɢ ɩɪɢ ɩɨɞɫɬɚɧɨɜɤɟ ɟɝɨ ɜ ɭɪɚɜɧɟɧɢɟ ɜɦɟɫɬɨ ɩɟɪɟɦɟɧɧɨɣ, ɨɧɨ ɨɛɪɚɳɚɟɬ ɭɪɚɜɧɟɧɢɟ ɜ ɬɨɠɞɟɫɬɜɨ

Ɉɫɧɨɜɧɚɹɬɟɨɪɟɦɚ ɚɥɝɟɛɪɵ ȼɫɹɤɨɟ ɭɪɚɜɧɟɧɢɟ ɧɟɧɭɥɟɜɨɣ ɫɬɟɩɟɧɢ ɢɦɟɟɬ ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɨɞɢɧ ɤɨɦɩɥɟɤɫɧɵɣ ɤɨɪɟɧɶ [12]

ɑɢɫɥɨ z0 ɧɚɡɵɜɚɟɬɫɹ ɤɨɪɧɟɦ ɫɬɟɩɟɧɢ k ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɟɫɥɢ ɷɬɨ ɭɪɚɜ ɧɟɧɢɟ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɜ ɜɢɞɟ

(z z0)k (bn k zn k ... b1z b0) 0.

Ɉɫɧɨɜɧɭɸɬɟɨɪɟɦɭɚɥɝɟɛɪɵɦɨɠɧɨɩɟɪɟɮɨɪɦɭɥɢɪɨɜɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚ ɡɨɦ ɥɸɛɨɟ ɭɪɚɜɧɟɧɢɟɫɬɟɩɟɧɢ n ɢɦɟɟɬ ɪɨɜɧɨ n ɤɨɦɩɥɟɤɫɧɵɯ ɤɨɪɧɟɣ ɫ ɭɱɟɬɨɦ ɢɯ ɤɪɚɬɧɨɫɬɢ

ɉɪɢɦɟɪ Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ

			z2 4 0.
ȿɝɨ ɤɨɪɧɹɦɢ ɹɜɥɹɸɬɫɹ ɤɜɚɞɪɚɬɧɵɟ ɤɨɪɧɢ ɢɡ ɱɢɫɥɚ 4 :
z 4 2(cos				Q2Qk	isin		Q2Qk	),
z 4 2(cos					isin			),
			2			2
ɩɨɞɫɬɚɜɥɹɹ k 0,1 ɩɨɥɭɱɚɟɦ ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ
		z1 2i, z2 2i.
Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ
			z2 2z 5 0.
ɉɨ ɮɨɪɦɭɥɟ ɞɥɹ ɤɨɪɧɟɣ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɟɦ
z 1		( 2 o 16) 1				( 2 o4i)
1,2	2		2
	2		2
	z1 1 2i, z2 1 2i.
Ɋɚɫɫɦɨɬɪɢɦ ɭɪɚɜɧɟɧɢɟ
	z2 (5 2i)z 5(1 i) 0.
ɉɨ ɮɨɪɦɭɥɟ ɞɥɹ ɤɨɪɧɟɣ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɟɦ
	z		1 ( 5 2i o 1).
	1,2		2
			2

z1 2 i, z2 3 i.

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