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Çà ôîðìóëîþ (13.1.9) îòðèìàºìî<br />
3 0+ 2kπ 0+ 2kπ<br />
1 =<br />
3<br />
cos0 + isin 0 = cos + isin .<br />
3 3<br />
Ïîêëàâøè k = 0, 1, 2, çíàõîäèìî òðè çíà÷åííÿ êîðåíÿ:<br />
x = cos0 + isin 0 = 1, x = cos(2 π / 3) + isin(2 π / 3) =− 1/ 2 + i 3 / 2,<br />
1 2<br />
x3 = cos(4 π / 3) + isin(4 π / 3) =−1/ 2 −i<br />
3 / 2.<br />
Ïðèêëàä 13.2. Çíàéòè âñ³ çíà÷åííÿ êâàäðàòíîãî êîðåíÿ ç<br />
ì³íóñ îäèíèö³.<br />
Ðîçâ’ÿçàííÿ. Çîáðàçèìî ì³íóñ îäèíèöþ â òðèãîíîìåòðè÷í³é<br />
ôîðì³<br />
–1 = cos π +isin π..<br />
Çà ôîðìóëîþ (13.1.9) îòðèìàºìî<br />
π+ 2kπ π+ 2kπ<br />
− 1 = cos π+ isin π = cos + isin .<br />
2 2<br />
Ïîêëàâøè k = 0, 1, çíàõîäèìî äâà çíà÷åííÿ êîðåíÿ:<br />
π π 3π 3π<br />
x1 = cos + isin = i, x2<br />
= cos( ) + isin( ) = −i.<br />
2 2 2 2<br />
13.1.4. Ðîçâ’ÿçóâàííÿ êâàäðàòíèõ ð³âíÿíü<br />
ç â³ä’ºìíèì äèñêðèì³íàíòîì<br />
Ïðè ðîçâ’ÿçóâàíí³ êâàäðàòíèõ ð³âíÿíü<br />
+ + =0<br />
2<br />
ax bx c<br />
ó âèïàäêó íåâ³ä’ºìíîãî äèñêðèì³íàíòà D (D = b 2 − 4ac)<br />
êîðèñòóþòüñÿ â³äîìîþ ôîðìóëîþ<br />
x<br />
− ±<br />
= b D<br />
2a<br />
1,2<br />
.<br />
ßêùî æ äèñêðèì³íàíò D â³ä’ºìíèé, òî êàæóòü, ùî íà<br />
ìíîæèí³ ä³éñíèõ ÷èñåë ð³âíÿííÿ íå ìຠðîçâ’ÿçê³â. ßê âæå<br />
ñêàçàíî, öåé ôàêò ïîâ’ÿçàíèé ç òèì, ùî çäîáóòòÿ êîðåíÿ ç<br />
â³ä’ºìíîãî ÷èñëà íå ìຠíà ìíîæèí³ ä³éñíèõ ÷èñåë ñåíñó.<br />
Íà ìíîæèí³ êîìïëåêñíèõ ÷èñåë òàêà îïåðàö³ÿ âæå ìຠñåíñ,<br />
îñê³ëüêè íà í³é âîíà ãàðàíòóº âèêîíóâàí³ñòü 䳿 äîáóâàííÿ<br />
êâàäðàòíîãî êîðåíÿ ç â³ä’ºìíîãî ÷èñëà. Ðîçãëÿíåìî öþ îïåðàö³þ.<br />
² ïî÷íåìî ¿¿ ç äîáóòòÿ êîðåíÿ êâàäðàòíîãî ç –1.<br />
Ïðèêëàä 13.2 ïîêàçàâ, ùî ³ñíóþòü äâà ³ ò³ëüêè äâà çíà÷åííÿ<br />
êîðåíÿ êâàäðàòíîãî ç –1, à ñàìå: ³ òà −³. Óìîâíî öå<br />
çàïèñóþòü òàê: − 1 = i, − − 1 =−i . Îá´ðóíòîâàí³ñòü òàêèõ ïîçíà÷åíü<br />
ìຠì³ñöå ³ âîíà ïîâ’ÿçàíà ç ìíîãîçíà÷íèìè ôóíêö³ÿìè<br />
êîìïëåêñíîãî çì³ííîãî. ×èòà÷, ÿêîãî ö³êàâèòü ñòðîã³ñòü<br />
ââåäåíèõ ïîíÿòü, ìîæå çàäîâîëüíèòèñü ìàòåð³àëîì,<br />
âèêëàäåíèì ó áóäü-ÿêîìó ïîñ³áíèêó ç òåî𳿠ôóíêö³é êîìïëåêñíîãî<br />
çì³ííîãî.<br />
Àíàëîã³÷íî ìîæíà ïîêàçàòè, ùî ³ñíóº äâà ³ ò³ëüêè äâà<br />
çíà÷åííÿ êîðåíÿ êâàäðàòíîãî ç –à, à>0, à ñàìå:<br />
−<br />
a ⋅i , äå ï³ä<br />
ÂÏÐÀÂÈ<br />
Âèêîíàòè àðèôìåòè÷í³ ä³¿:<br />
a ðîçóì³þòü àðèôìåòè÷íèé êîð³íü.<br />
13.1. (4+7i) + (1+5i); 13.2. (5-7i) –(3-4i);<br />
13.3. (5+7i)(5-7i); 13.4.<br />
13.5.<br />
13.7. 1 + i 2<br />
1-i 2<br />
⎛<br />
⎞<br />
⎛ 1 3 1 3<br />
i<br />
⎞ ⎛ i<br />
⎞<br />
⎜<br />
+ + −<br />
2 2 ⎟ ⎜ 2 2 ⎟<br />
⎝ ⎠ ⎝ ⎠ ;<br />
⎛ ⎞⎛ ⎞<br />
⎜ ⎟⎜ ⎟<br />
⎝ ⎠⎝ ⎠ ;<br />
⎛1 3 ⎞ 1 3<br />
⎜ + i i<br />
2 ⎟− −<br />
⎝2 ⎠<br />
⎜2 2 ⎟<br />
⎝ ⎠ ; 13.6. 1 3 1 3<br />
+ i −i<br />
2 2 2 2<br />
13.8. -2 3 + i<br />
1+<br />
2 3i<br />
13.9.<br />
1 3<br />
+ i<br />
2 2<br />
1 3 .<br />
− i<br />
2 2<br />
a ⋅i òà<br />
Çíàéòè ìîäóë³ ³ ãîëîâí³ çíà÷åííÿ äàíèõ êîìïëåêñíèõ<br />
÷èñåë:<br />
13.10. z =3+4i; 13.11. z = 1+ 3i; 13.12. z =1+i;<br />
13.13. z =–1–i; 13.14. z = i; 13.15. z =–i;<br />
486 487
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