- në përgjithësi për in, n
vlen:
1, kur n 4k
i, kur n 4k+ 1 -1, kur n 4k+2 (32) -i, kur n 4k+3.
3.4. RRËNJËZIMI DHE LOGATIRMIMI I NUMRIT KOMPLEKS
- P ë r k u f i z i m i 3.4.1. - Rrënja
e numrit kompleks quhet numri kompleks i tillë që fuqia e tij është e barabartë me , pra:
![{\displaystyle \scriptstyle \mathrm {\sqrt[{\mathrm {n} }]{\mathrm {z} }} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/beee04709866412744fd178031ceafc1be3826bd) w wn z, n![{\displaystyle \scriptstyle \in }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a4e4b18f88a540b881dab023807160f159f8a9) . (...33)
- Kur në barazinë e fundit zëvendësohen format trigonometrike të numrave kompleksë z, w përftohet:
n(cos n + i sin n ) r (cos + i sin ),
- nga del:
n r dhe n![{\displaystyle {\theta }\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc2794d03179d3cd580a063ef0c42f805338922) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) +2k ,
- respektivisht
![{\displaystyle {\rho }\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09e23b0d125a66d2e5d80c41d125505111d37681) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) vlera aritmetike e rrënjës,
![{\displaystyle {\theta }\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc2794d03179d3cd580a063ef0c42f805338922) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) k 0, 1, 2,..., n- 1.
- Pra:
![{\displaystyle \scriptstyle \mathrm {\sqrt[{n}]{\mathrm {r(\cos \varphi +i\ \sin \varphi )} }} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/baca0b61248e4816fda9a52f52a5007fbbffab44) ![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) ![{\displaystyle \scriptstyle \mathrm {\sqrt[{\mathrm {n} }]{\mathrm {r} }} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa4be98b953ee3b6e7950a024262b7947d83d6b6) cos +i sin![{\displaystyle \textstyle \mathrm {\frac {\varphi +2k\pi }{n}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0936d3a401f61a3b1a9ae45c9fc7b78747e6f3a3) , (...33a)
- ku k
0,1,2,...,n-1.
- Nga kjo formulë shihet se rrënja n e numrit kompleks ka gjithsej n vlera të ndryshme.
- S h e m b u l l i 4. - Të njehsohet
.
- Z g j i d h j e : Në bazë të formulës (33a) del:
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zk![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc)
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![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc)
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![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) cos +i sin![{\displaystyle \textstyle \mathrm {\frac {-45^{\circ }+2k\pi }{5}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9d585150e113f1231e52e730ce20257712debcb) , k 0, 1, 2, 3, 4.
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Për k 0:
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z0![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) [cos (-9°)+i sin (-9°)]![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) (0,98769-0,15643 • i);
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Për k 1:
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z1![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) (cos 63° +i sin 63°)![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) (0,45399+0,89101 • i).
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Për k 2 : z2![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) (-1+i); k 3: z3![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) (0,89101 +0,45399 . i); k 4: z4![{\displaystyle \scriptstyle {=}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d0a664fb934fdecbb413b38ad8f5c7abf1932cc) (0,15643 - 0,98769 • i).
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