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òî<br />
z<br />
z r (cos ϕ + isin ϕ ) r<br />
(cos( ) sin( )).<br />
1 1 1 1 1<br />
= = = ϕ1 −ϕ<br />
2<br />
+ i ϕ1 −ϕ2<br />
z2 r2(cos ϕ<br />
2<br />
+ isin ϕ2)<br />
r2<br />
(13.1.7)<br />
Äëÿ ïåðåâ³ðêè ö³º¿ ð³âíîñò³ äîñòàòíüî ïîìíîæèòè ä³ëüíèê<br />
íà ÷àñòêó.<br />
Òàêèì ÷èíîì, ìîäóëü ÷àñòêè äâîõ êîìïëåêñíèõ ÷èñåë<br />
äîð³âíþº ÷àñòö³ ìîäóë³â ä³ëåíîãî ³ ä³ëüíèêà; àðãóìåíò ÷àñòêè<br />
äîð³âíþº ð³çíèö³ àðãóìåíò³â ä³ëåíîãî ³ ä³ëüíèêà.<br />
5. Çâåäåííÿ â ñòåï³íü. Ç ôîðìóëè (13.1.5) âèïëèâàº, ùî<br />
ÿêùî n — ö³ëå äîäàòíå ÷èñëî, òî<br />
n n<br />
((cos r ϕ+ isin ϕ )) = r (cosnϕ+ isin n ϕ)<br />
. (13.1.8)<br />
Öÿ ôîðìóëà íàçèâàºòüñÿ ôîðìóëîþ Ìóàâðà 1 .<br />
Ðîçãëÿíåìî îäíå ç çàñòîñóâàíü ôîðìóëè Ìóàâðà. Ïîêëàäåìî<br />
â ôîðìóë³ (1.3.10) r = 1, òîä³ îòðèìàºìî<br />
n<br />
(cos ϕ+ isin ϕ ) = cos nϕ+ isin nϕ.<br />
Ðîçêëàäàþ÷è ë³âó ÷àñòèíó çà ôîðìóëîþ á³íîìà Íüþòîíà<br />
(âîíà áóäå äîâåäåíà ï³çí³øå) ³ äîð³âíþþ÷è ä³éñí³ é óÿâí³<br />
÷àñòèíè, ìîæíà âèðàçèòè sin nϕ ³ cosnϕ ÷åðåç ñòåïåí³ sin ϕ<br />
³ cos ϕ. Íàïðèêëàä, ó âèïàäêó n = 3 îòðèìàºìî<br />
3 2 2 3<br />
cos ϕ+ i3cos ϕsin ϕ−3cos ϕsin ϕ−isin ϕ= cos3ϕ+ isin 3ϕ⇒<br />
⇒ ϕ− ϕ ϕ= ϕ − ϕ+ ϕ ϕ= ϕ<br />
3 2 3 2<br />
cos 3cos sin cos3 , sin 3cos sin sin 3 .<br />
6. Äîáóâàííÿ êîðåíÿ. Êîðåíåì n-ãî ñòåïåíÿ ç êîìïëåêñíîãî<br />
÷èñëà íàçèâàºòüñÿ òàêå êîìïëåêñíå ÷èñëî, n-é ñòåï³íü<br />
ÿêîãî äîð³âíþº ï³äêîðåíåâîìó ÷èñëó, òîáòî<br />
ÿêùî<br />
n<br />
r(cosϕ+ isin ϕ ) =ρ(cosψ+ isin ψ),<br />
n<br />
ρ (cos nψ+ isin nψ ) = r(cos ϕ+ isin ϕ).<br />
Îñê³ëüêè ó ð³âíèõ êîìïëåêñíèõ ÷èñåë ìîäóë³ ïîâèíí³<br />
áóòè ð³âí³, à àðãóìåíòè ìîæóòü â³äð³çíÿòèñÿ íà ÷èñëî êðàòíå<br />
2π, òî<br />
n n ϕ+ 2kπ<br />
ρ = r, nψ=ϕ+ 2 kπ⇒ρ= r, ψ= ,<br />
n<br />
äå k — áóäü-ÿêå ö³ëå ÷èñëî; n r — àðèôìåòè÷íå çíà÷åííÿ<br />
êîðåíÿ. Îòæå,<br />
n ϕ+ 2kπ ϕ+ 2kπ<br />
n<br />
r(cosϕ+ isin ϕ ) = r(cos + isin ). (13.1.9)<br />
n<br />
n<br />
Íàäàþ÷è k çíà÷åííÿ 0, 1, 2,..., n-1, îòðèìàºìî n ð³çíèõ<br />
çíà÷åíü êîðåíÿ. Äëÿ ³íøèõ çíà÷åíü, íàïðèêëàä äëÿ k = n,<br />
n+1,..., àðãóìåíòè áóäóòü â³äð³çíÿòèñÿ â³ä îòðèìàíèõ íà<br />
÷èñëî, êðàòíå 2π ³, îòæå, áóäóòü îòðèìàí³ çíà÷åííÿ êîðåíÿ,<br />
ùî çá³ãàþòüñÿ ç ðîçãëÿíóòèìè. Òàêèì ÷èíîì, êîð³íü n-ãî<br />
ñòåïåíÿ ç êîìïëåêñíîãî ÷èñëà ìຠn ð³çíèõ çíà÷åíü.<br />
Êîð³íü n-ãî ñòåïåíÿ ç ä³éñíîãî ÷èñëà a, â³äì³ííîãî â³ä<br />
íóëÿ, òàêîæ ìຠn çíà÷åíü, òîìó ùî ä³éñíå ÷èñëî a º îêðåìèì<br />
âèïàäêîì êîìïëåêñíîãî ³ ìîæå áóòè çîáðàæåíî â<br />
òðèãîíîìåòðè÷í³é ôîðì³:<br />
⎧ ⎪ a (cos0 + isin0), a≥0,<br />
a = ⎨<br />
⎪⎩ a (cosπ+ isin π ), a<<br />
0.<br />
Íàïðèê³íö³ öüîãî ïóíêòó íàâåäåìî ùå îäèí çàïèñ êîìïëåêñíèõ<br />
÷èñåë.<br />
Ó â³äïîâ³äíîñò³ äî ôîðìóëè Åéëåðà (äèâ. 12.5.12) ìàºìî<br />
i<br />
e<br />
ϕ = cos ϕ+ isin<br />
ϕ. (13.1.10)<br />
Òîä³ êîìïëåêñíå ÷èñëî z ìîæå áóòè çîáðàæåíå ó âèãëÿä³<br />
z<br />
i<br />
= z e ϕ .<br />
Ïðèêëàä 13.1. Çíàéòè âñ³ çíà÷åííÿ êóá³÷íîãî êîðåíÿ ç<br />
îäèíèö³.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Çîáðàçèìî îäèíèöþ â òðèãîíîìåòðè÷í³é<br />
ôîðì³<br />
1 = cos 0 +isin 0.<br />
1<br />
Ìóàâð Àáðàõàì (1667 – 1754) — àíãë³éñüêèé ìàòåìàòèê.<br />
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