Hipi Zhdripi i Matematikës/1077

vlera absolute, kurse këndi polar $\scriptstyle {\varphi }\,\!$ quhet argumenti i numrit kompleks z. Ngadonjëherë moduli i numrit kompleks z shënohet me mod (z) ose $\scriptstyle \mid$z$\scriptstyle \mid$, kurse argumenti me arg (z). Nga figura 3.2. shihet se moduli dhe argumenti i numrit kompleks z$\scriptstyle {=}$x+iy mund të përcaktohen me formulat: r$\scriptstyle {=}$$\scriptstyle {\mathsf {\sqrt {x^{2}+y^{2}}}}$, $\scriptstyle {\varphi }\,\!$$\scriptstyle {=}$arc tg $\textstyle \mathrm {\frac {y}{x}}$, (...12) prandaj ($\scriptstyle {\forall }$z$\scriptstyle \in$$\scriptstyle \mathbb {C}$)z$\scriptstyle {\neq }$0$\scriptstyle \Leftrightarrow$ $\scriptstyle \mid$z$\scriptstyle \mid$>0. (...13)        P.sh. moduli i numrit kompleks z$\scriptstyle {=}$-$\scriptstyle {\mathsf {\sqrt {3}}}$ +i është: r$\scriptstyle {=}$$\scriptstyle {\mathsf {\sqrt {(-{\sqrt {3}})^{2}+1^{2}}}}$$\scriptstyle {=}$2, argumenti i tij është: $\scriptstyle {\varphi }\,\!$$\scriptstyle {=}$ arc tg $\textstyle \mathrm {\frac {1}{-{\sqrt {3}}}}$ $\scriptstyle {=}$ arc tg $\textstyle \mathrm {\frac {-{\sqrt {3}}}{3}}$ $\scriptstyle {\Rightarrow }$ ${\pi }\,\!$-$\scriptstyle {\varphi }\,\!$ $\scriptstyle {=}$ arc tg $\textstyle \mathrm {\frac {\sqrt {3}}{3}}$ $\scriptstyle {=}$ $\textstyle \mathrm {\frac {\pi }{6}}$ $\scriptstyle {\Rightarrow }$ $\textstyle \mathrm {\frac {5\pi }{6}}$ nga del se forma trigonometrike e tij është: z$\scriptstyle {=}$2 ${\bigl (}$cos $\textstyle \mathrm {\frac {5\pi }{6}}$ +i sin $\textstyle \mathrm {\frac {5\pi }{6}}$ ${\bigl )}$. Në raste të posaçme kemi:         mod (x,0) ${\bigg \{}$ x, kur x > 0 -x kur x < 0 arg (x,0) ${\bigg \{}$ 0, kur x > 0 ${\pi }\,\!$ kur x < 0        dhe mod (y,0) ${\bigg \{}$ y, kur y > 0 -y kur y < 0 arg (y,0) ${\bigg \{}$ ${\pi }\,\!$/2, kur y > 0 - ${\pi }\,\!$/2 kur y < 0        Kur në formën trigonometrike të numrit kompleks z $\scriptstyle {=}$ r (cos $\scriptstyle {\varphi }\,\!$ + i sin $\scriptstyle {\varphi }\,\!$) zëvendësohet formula e Eulerit ${e}\,\!$± i$\scriptstyle {\varphi }\,\!$$\scriptstyle {=}$cos $\scriptstyle {\varphi }\,\!$±i sin$\scriptstyle {\varphi }\,\!$ (...14) numri kompleks z transformohet në formën z $\scriptstyle {=}$ r${e}\,\!$i$\scriptstyle {\varphi }\,\!$ (...15) që quhet forma eksponenciale ose forma e Eulerit të numrit kompleks z.

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