Ad argvmentvm transi
Topos Uranos
Latine

Forma polaris nvmerorvm complexorvm

Svmmarivm

Haec lectio omnem nvmervm complexvm non nvllvm per modvlvm svvm et argvmentvm, angvlvm polarem pvncti qvod evm repraesentat, scribit, et ita formam eivs polarem sive trigonometricam obtinet, z=r(cos⁡θ+isin⁡θ)z = r\left( \cos \theta + i \sin \theta \right). Omnia argvmenta nvmeri complexi describit, argvmentvm principale definit idqve arcv tangente pro qvadrante correcto compvtat, et demonstrat qvando dvae formae polares evndem nvmervm repraesentent. Formvlis additionis et svbtractionis theorema centrale demonstrat: cvm dvo complexi mvltiplicantvr, modvli eorvm mvltiplicantvr et argvmenta addvntvr, et cvm dividvntvr, modvli dividvntvr et argvmenta svbtrahvntvr; ex eo argvmentvm conivgati et argvmentvm inversi dedvcvntvr, et monetvr argvmentvm principale prodvcti svmmam argvmentorvm principalivm esse tantvm nisi integra conversione. Postremo prodvctvm per nvmervm complexvm vt conversionem cvm dilatatione compositam interpretatvr, qvae conversionem qvadrantis prodvcti per ii generalem reddit, et demonstrat triangvla verticvm 00, 11, ww et 00, zz, zwzw similia esse.

Proposita discendi

  1. Nvmervm complexvm non nvllvm forma polari scribere, argvmentvm eivs principale pro qvadrante compvtare et diivdicare qvando dvae formae polares evndem nvmervm repraesentent.
  2. Formvlis additionis et svbtractionis demonstrare in prodvcto modvlos mvltiplicari et argvmenta addi, et regvlas qvotientis, conivgati et inversi dedvcere.
  3. Prodvctvm per nvmervm complexvm vt conversionem cvm dilatatione compositam interpretari, et similitvdinem triangvlorvm verticvm 00, 11, ww et 00, zz, zwzw demonstrare.
  4. Formam polarem adhibere ad valores exactos rationvm trigonometricarvm, ad identitates inter arcvs tangentivm, ad aeqvationes cvm zz eivsqve conivgato et ad locos geometricos per argvmenta definitos.

Forma polaris nvmeri complexi

In lectione cvrsvs Algebra et Geometria I de nvmeris complexis qvisqve complexvs z=a+biz = a + bi vt pvnctvm (a,b)(a, b) plani repraesentatvs est, modvlvs eivs ∣z∣\abs{z} vt distantia illivs pvncti ab origine definitvs est, et demonstratvm est prodvctvm per ii qvodqve pvnctvm qvadrante convertere. Restabat prodvctvm per complexvm qvemlibet, cvivs effectvs in forma binomiali non legitvr: pars realis ipsivs (a+bi)(c+di)(a + bi)(c + di) est ac−bdac - bd, expressio qvae nvllam geometriam patefacit. Atqvi pvnctvm plani etiam per distantiam svam ab origine et per directionem qva ex ea conspicitvr nominari potest, vt in lectione de coordinatis polaribvs. Qvod si ita est in pvnctis, ita est in complexis; et in illa altera descriptione, vt demonstrabitvr, prodvctvm perspicvvm fit.

Sit z=a+biz = a + bi complexvs non nvllvs. Argvmentvm ipsivs zz est omnis nvmervs realis θ\theta talis vt a=∣z∣cos⁡θa = \abs{z} \cos \theta et b=∣z∣sin⁡θb = \abs{z} \sin \theta. Si r=∣z∣r = \abs{z} et θ\theta est argvmentvm ipsivs zz, expressio infra scripta est forma polaris, sive forma trigonometrica, ipsivs zz.
z=r(cos⁡θ+isin⁡θ)z = r\left( \cos \theta + i \sin \theta \right)

Omnis complexvs non nvllvs argvmenta habet. Etenim ex theoremate de transitv a coordinatis Cartesianis ad polares, in lectione de coordinatis polaribvs demonstrato, pvnctvm (a,b)(a, b), a polo diversvm, repraesentationem polarem (r,θ)(r, \theta) cvm r>0r > 0 habet, et radivs eivs est r=a2+b2r = \sqrt{a^2 + b^2}, id est modvlvs ipsivs zz; aeqvalitates a=rcos⁡θa = r \cos \theta et b=rsin⁡θb = r \sin \theta tvm dicvnt angvlvm polarem θ\theta argvmentvm esse ipsivs zz. Argvmentvm ipsivs zz est igitvr angvlvs qvem cvm semiaxe reali positivo facit segmentvm qvod originem cvm zz ivngit, in radianis mensvs. Nihilvm avtem argvmento caret: aeqvalitas 0=0(cos⁡θ+isin⁡θ)0 = 0\left( \cos \theta + i \sin \theta \right) valet pro omni θ\theta, et nvllvs angvlvs id distingvit. Ad qvodlibet argvmentorvm ipsivs zz designandvm scribitvr arg⁡z\arg z, qvod legitvr «argvmentvm ipsivs zz».

Theorema qvod seqvitvr dicit modvlvm et argvmentvm, dissimiliter atqve coordinatas polares radii negativi, complexvm sine ambigvitate determinare, nisi integris conversionibvs. Ex eo omnis compvtatio cvm formis polaribvs pendet: ad demonstrandvm dvos complexos aeqvales esse satis erit modvlos eorvm et argvmenta comparare.

TheoremaAeqvalitas in forma polari

Sint rr, r1r_1 et r2r_2 reales positivi, et θ\theta, θ1\theta_1 et θ2\theta_2 nvmeri reales. Complexvs r(cos⁡θ+isin⁡θ)r\left( \cos \theta + i \sin \theta \right) modvlvm rr et argvmentvm θ\theta habet. Praeterea complexi r1(cos⁡θ1+isin⁡θ1)r_1\left( \cos \theta_1 + i \sin \theta_1 \right) et r2(cos⁡θ2+isin⁡θ2)r_2\left( \cos \theta_2 + i \sin \theta_2 \right) aeqvales svnt si et solvm si r1=r2r_1 = r_2 et θ1−θ2=2kπ\theta_1 - \theta_2 = 2k\pi pro aliqvo integro kk. Adhibentvr identitas Pythagorica, periodicitas sinvs et cosinvs et descriptio angvlorvm coterminalivm ex lectione de angvlis eorvmqve mensvra: dvo angvli idem pvnctvm circvmferentiae vnitariae dant exacte cvm mvltiplo integro ipsivs 2π2\pi differvnt.

Demonstratio

  1. ∣z∣2=r2cos⁡2θ+r2sin⁡2θ=r2\abs{z}^2 = \resaltar{r^2 \cos^2 \theta + r^2 \sin^2 \theta} = \resaltar{r^2}

    Sit z=r(cos⁡θ+isin⁡θ)z = r\left( \cos \theta + i \sin \theta \right), cvm r>0r > 0. Pars eivs realis est rcos⁡θr \cos \theta et pars imaginaria rsin⁡θr \sin \theta; ex identitate Pythagorica qvadratvm modvli eivs est r2r^2.

  2. ∣z∣=r,rcos⁡θ=∣z∣cos⁡θ,rsin⁡θ=∣z∣sin⁡θ\dato{m}{\resaltar{\abs{z} = r}}, \qquad r \cos \theta = \resaltar{\abs{z} \cos \theta}, \qquad r \sin \theta = \abs{z} \sin \theta

    Nvmeri ∣z∣\abs{z} et rr positivi svnt et idem qvadratvm habent; itaqve aeqvales svnt, et pars realis et imaginaria ipsivs zz eae svnt qvas definitio postvlat: θ\theta est argvmentvm ipsivs zz. Omnis igitvr expressio hvivs formae cvm r>0r > 0 est forma polaris.

  3. r1(cos⁡θ1+isin⁡θ1)=r2(cos⁡θ2+isin⁡θ2)→r1=r2r_1 \left( \cos \theta_1 + i \sin \theta_1 \right) = r_2 \left( \cos \theta_2 + i \sin \theta_2 \right) \rightarrow \dato{r}{\resaltar{r_1 = r_2}}

    Si complexi modvli r1r_1 et r2r_2 aeqvales svnt, evndem modvlvm habent, et ex eo qvod svpra ostensvm est modvli eorvm svnt r1r_1 et r2r_2.

  4. cos⁡θ1=cos⁡θ2∧sin⁡θ1=sin⁡θ2↔θ1−θ2=2kπ\resaltar{\cos \theta_1 = \cos \theta_2 \wedge \sin \theta_1 = \sin \theta_2} \leftrightarrow \resaltar{\theta_1 - \theta_2 = 2k\pi}

    Partem realem cvm parte reali et imaginariam cvm imaginaria aeqvamvs et per r1=r2r_1 = r_2 dividimvs, qvi non evanescit: angvli θ1\theta_1 et θ2\theta_2 idem pvnctvm circvmferentiae vnitariae dant, ita vt coterminales sint.

  5. cos⁡(θ2+2kπ)=cos⁡θ2,sin⁡(θ2+2kπ)=sin⁡θ2\cos(\resaltar{\theta_2 + 2k\pi}) = \resaltar{\cos \theta_2}, \qquad \sin(\theta_2 + 2k\pi) = \sin \theta_2

    Vicissim, si r1=r2r_1 = r_2 et θ1=θ2+2kπ\theta_1 = \theta_2 + 2k\pi, ex periodicitate sinvs et cosinvs dvae expressiones eandem partem realem et eandem imaginariam habent, et complexi aeqvales svnt.

Inde seqvitvr, si θ\theta est argvmentvm ipsivs zz, argvmenta ipsivs zz exacte esse nvmeros θ+2kπ\theta + 2k\pi, cvm kk integro: dvo argvmenta eivsdem complexi semper integris conversionibvs differvnt. Aeqvalitas arg⁡z=θ\arg z = \theta igitvr significat θ\theta vnvm esse ex argvmentis ipsivs zz, nec cetera figit. Praeterea animadvertendvm est theorema r>0r > 0 postvlare: expressio −2(cos⁡π3+isin⁡π3)-2\left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right) non est forma polaris, qvia −2-2 non est modvlvs; cvm −cos⁡θ=cos⁡(θ+π)-\cos \theta = \cos(\theta + \pi) et −sin⁡θ=sin⁡(θ+π)-\sin \theta = \sin(\theta + \pi), idem nvmervs forma polari scribitvr vt 2(cos⁡4π3+isin⁡4π3)2\left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right).

Argvmentvm principale

Inter argvmenta complexi non nvlli exacte vnvm est in intervallo (−π,π]\ioc{-\pi}{\pi}, qvia illvd intervallvm longitvdinem 2π2\pi habet et vnicvm repraesentantem cvivsqve familiae θ+2kπ\theta + 2k\pi continet. Id argvmentvm argvmentvm principale ipsivs zz appellatvr, scribitvr Arg⁡z\operatorname{Arg} z et legitvr «argvmentvm principale ipsivs zz». Ex theoremate lectionis de coordinatis polaribvs, cvm aa et bb mvnere ipsorvm xx et yy fvngentibvs, compvtatvr arcv tangente lectionis de fvnctionibvs trigonometricis inversis, qvi primvm et qvartvm tantvm qvadrantem videt et in ceteris dvobvs corrigendvs est:

Arg⁡z={arctan⁡baa>0arctan⁡ba+πa<0∧b≥0arctan⁡ba−πa<0∧b<0π2a=0∧b>0−π2a=0∧b<0\operatorname{Arg} z = \begin{cases} \arctan \frac{b}{a} & a > 0 \\ \arctan \frac{b}{a} + \pi & a < 0 \wedge b \geq 0 \\ \arctan \frac{b}{a} - \pi & a < 0 \wedge b < 0 \\ \frac{\pi}{2} & a = 0 \wedge b > 0 \\ -\frac{\pi}{2} & a = 0 \wedge b < 0 \end{cases}
ExemplvmForma polaris complexi secvndi qvadrantis

Complexvm z=−3+iz = -\sqrt{3} + i forma polari scribimvs et omnia eivs argvmenta describimvs. Figvra compvtationem in plano complexo seqvitvr.

Demonstratio

  1. ∣z∣=(−3)2+12=4=2\abs{z} = \sqrt{\resaltar{\left( -\sqrt{3} \right)^2 + 1^2}} = \sqrt{4} = \dato{m}{\resaltar{2}}

    Modvlvs est distantia pvncti (−3,1)\left( -\sqrt{3}, 1 \right) ab origine.

  2. arctan⁡1−3=arctan⁡(−33)=−π6\arctan \frac{1}{-\sqrt{3}} = \resaltar{\arctan\left( -\frac{\sqrt{3}}{3} \right)} = \dato{a}{\resaltar{-\frac{\pi}{6}}}

    Arcvs tangens qvotientis partis imaginariae per partem realem angvlvm qvarti qvadrantis dat: est argvmentvm pvncti oppositi, 3−i\sqrt{3} - i, non ipsivs zz.

  3. Arg⁡z=−π6+π=5π6\operatorname{Arg} z = \resaltar{-\frac{\pi}{6} + \pi} = \dato{t}{\resaltar{\frac{5\pi}{6}}}

    Cvm pars realis negativa et imaginaria positiva sit, pvnctvm in secvndo qvadrante iacet, et formvla π\pi addit.

  4. z=2(cos⁡5π6+isin⁡5π6)=2(−32+12i)=−3+iz = 2\left( \resaltar{\cos \frac{5\pi}{6}} + i \sin \frac{5\pi}{6} \right) = 2\left( \resaltar{-\frac{\sqrt{3}}{2}} + \frac{1}{2} i \right) = -\sqrt{3} + i

    Formam polarem valoribvs cosinvs et sinvs ipsivs 5π6\frac{5\pi}{6} confirmamvs, qvi ex valoribvs ipsivs π6\frac{\pi}{6} per symmetriam angvli svpplementaris obtinentvr.

  5. arg⁡z=5π6+2kπ,k∈Z\arg z = \resaltar{\frac{5\pi}{6} + 2k\pi}, \qquad k \in \mathbb{Z}

    Cetera argvmenta a principali integris conversionibvs differvnt: exempli gratia, −7π6-\frac{7\pi}{6} et 17π6\frac{17\pi}{6} qvoqve argvmenta svnt ipsivs zz, sed in (−π,π]\ioc{-\pi}{\pi} non iacent.

Complexi in axibvs siti formas polares habent qvas statim agnoscere convenit. Reales positivi argvmentvm principale 00 habent, negativi π\pi; imaginarii pvri bibi cvm b>0b > 0 argvmentvm principale π2\frac{\pi}{2} habent, et cvm b<0b < 0, −π2-\frac{\pi}{2}. Ita, exempli gratia:

−1=cos⁡π+isin⁡π,i=cos⁡π2+isin⁡π2,−2i=2(cos⁡(−π2)+isin⁡(−π2))-1 = \cos \pi + i \sin \pi, \qquad i = \cos \frac{\pi}{2} + i \sin \frac{\pi}{2}, \qquad -2i = 2\left( \cos\left( -\frac{\pi}{2} \right) + i \sin\left( -\frac{\pi}{2} \right) \right)

Intervallvm (−π,π]\ioc{-\pi}{\pi} est conventio, eadem atqve in lectione de coordinatis polaribvs; alii libri [0,2π)\ico{0}{2\pi} adhibent. Qvod a conventione non pendet est copia omnivm argvmentorvm, et ideo theoremata hvivs lectionis argvmentis qvibvslibet envntiantvr, nisi expresse contrarivm dicitvr.