Ad argvmentvm transi
Topos Uranos
Latine

Formvla Moivreana et potentiae nvmerorvm complexorvm

Svmmarivm

Haec lectio formvlam Moivreanam demonstrat, secvndvm qvam potentia exponentis integri nn nvmeri complexi modvli 11 et argvmenti θ\theta argvmentvm nθn\theta habet, eamqve qvinqve modis adhibet. Potentias exponentis magni compvtat prodvctis non evolvtis, et decernit qvando potentia realis sit vel 11 valeat; formvlas angvlorvm mvltiplicivm recvperat formvlam cvm binomio Newtoniano comparando; potentias cosinvs et sinvs vt svmmas cosinvvm angvlorvm mvltiplicivm exprimit, qvae operatio linearizatio appellatvr; svmmas sinvvm et cosinvvm in progressione arithmetica vt svmmam geometricam complexorvm compvtat; deniqve svmmas coefficientivm binomialivm vt partem realem et imaginariam potentiarvm nvmeri 1+i1 + i obtinet.

Proposita discendi

  1. Formvlam Moivreanam pro omni exponente integro demonstrare et potentias complexi geometrice interpretari.
  2. Potentias complexorvm ope formae polaris compvtare et determinare qvando reales sint vel 11 aeqvales.
  3. Formvlas angvlorvm mvltiplicivm dedvcere et potentias sinvs et cosinvs linearizare.
  4. Svmmas trigonometricas et svmmas coefficientivm binomialivm compvtare vt partem realem et imaginariam svmmarvm et potentiarvm complexorvm.

Formvla Moivreana

In lectione de forma polari nvmerorvm complexorvm omnis complexvs non nvllvs scriptvs est vt z=r(cos⁡θ+isin⁡θ)z = r \left( \cos \theta + i \sin \theta \right), vbi r=∣z∣r = \abs{z} est eivs modvlvs et θ\theta vnvm ex eivs argvmentis, et demonstratvm est, ope formvlarvm sinvs et cosinvs svmmae, in prodvcto modvlos mvltiplicari et argvmenta addi. Pro dvobvs complexis modvli 11 illvd theorema prodvcti sic scribitvr:

(cos⁡α+isin⁡α)(cos⁡β+isin⁡β)=cos⁡(α+β)+isin⁡(α+β)\left( \cos \alpha + i \sin \alpha \right) \left( \cos \beta + i \sin \beta \right) = \cos(\alpha + \beta) + i \sin(\alpha + \beta)

Si dvo factores aeqvales svnt, prodvctvm est qvadratvm, argvmenti 2θ2\theta; si itervm per evndem factorem mvltiplicatvr, cvbvs fit, argvmenti 3θ3\theta. Prodvcto iterato, argvmentvm sibi ipsi totiens additvr qvot svnt factores, ita vt ad potentiam elevare eo redeat vt angvlvs per integrvm mvltiplicetvr. Hoc factvm, qvo Abraham de Moivre forma aeqvivalenti inevnte saecvlo dvodevicesimo vsvs est et qvod Evlervs anno 1748 forma hodierna scripsit, hvivs lectionis materia est.

TheoremaFormvla Moivreana

Pro omni nvmero reali θ\theta et omni integro nn aeqvalitas infra scripta valet.

(cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)\left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta)

Demonstratio tres casvs tractat: exponentem 00, exponentes natvrales, per indvctionem, et exponentes negativos, ope inversi. Adhibentvr principivm indvctionis forma lectionis cvrsvs Algebra et Geometria I de svmmationibvs, progressionibvs et indvctione, et potentiae exponentis integri ex lectione de nvmeris complexis, vbi z0=1z^0 = 1 et z−m=1zmz^{-m} = \frac{1}{z^m} statvta svnt. Figvra demonstrationem in circvlo radii 11 seqvitvr, angvlo θ\theta parvo et n=3n = 3.

Demonstratio

  1. (cos⁡θ+isin⁡θ)0=1=cos⁡0+isin⁡0\left( \cos \theta + i \sin \theta \right)^0 = \resaltar{1} = \resaltar{\cos 0 + i \sin 0}

    Pro n=0n = 0 membrvm privs ex definitione 11 valet, et secvndvm qvoqve, qvia cos⁡0=1\cos 0 = 1 et sin⁡0=0\sin 0 = 0.

  2. (cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)\dato{h}{\left( \cos \theta + i \sin \theta \right)^n = \cos(n\theta) + i \sin(n\theta)}

    Pro n=1n = 1 aeqvalitas manifesta est. Hypothesis indvctionis: ponimvs eam valere pro aliqvo natvrali nn.

  3. (cos⁡θ+isin⁡θ)n+1=(cos⁡θ+isin⁡θ)n(cos⁡θ+isin⁡θ)=(cos⁡(nθ)+isin⁡(nθ))(cos⁡θ+isin⁡θ)\left( \cos \theta + i \sin \theta \right)^{n + 1} = \resaltar{\left( \cos \theta + i \sin \theta \right)^n} \left( \cos \theta + i \sin \theta \right) = \resaltar{\left( \cos(n\theta) + i \sin(n\theta) \right)} \left( \cos \theta + i \sin \theta \right)

    Potentia qvae deinde venit est prodvctvm eivs qvae antecedit per basim, et haec ex hypothesi indvctionis svbstitvitvr.

  4. (cos⁡θ+isin⁡θ)n+1=(cos⁡(nθ)cos⁡θ−sin⁡(nθ)sin⁡θ)+i(sin⁡(nθ)cos⁡θ+cos⁡(nθ)sin⁡θ)=cos⁡((n+1)θ)+isin⁡((n+1)θ)\left( \cos \theta + i \sin \theta \right)^{n + 1} = \resaltar{\left( \cos(n\theta) \cos \theta - \sin(n\theta) \sin \theta \right)} + i \resaltar{\left( \sin(n\theta) \cos \theta + \cos(n\theta) \sin \theta \right)} = \dato{p}{\cos((n + 1)\theta) + i \sin((n + 1)\theta)}

    Prodvcto evolvto, pars realis et pars imaginaria svnt membra secvnda formvlarvm cosinvs et sinvs svmmae, ex lectione de formvlis additionis et svbtractionis: hoc est theorema de prodvcto et qvotiente in forma polari. Formvla pro n+1n + 1 obtinetvr, et ex principio indvctionis pro omni natvrali valet.

  5. (cos⁡θ+isin⁡θ)−1=cos⁡(−θ)+isin⁡(−θ)=cos⁡θ−isin⁡θ\left( \cos \theta + i \sin \theta \right)^{-1} = \dato{i}{\resaltar{\cos(-\theta) + i \sin(-\theta)}} = \resaltar{\cos \theta - i \sin \theta}

    Ex theoremate prodvcti cos⁡θ+isin⁡θ\cos \theta + i \sin \theta per cos⁡(−θ)+isin⁡(−θ)\cos(-\theta) + i \sin(-\theta) mvltiplicatvm dat cos⁡0+isin⁡0=1\cos 0 + i \sin 0 = 1: secvndvs igitvr factor est inversvm primi. Cvm cosinvs par sit et sinvs impar, inversvm idem est ac nvmervs conivgatvs.

  6. (cos⁡θ+isin⁡θ)−m=(cos⁡(−θ)+isin⁡(−θ))m=cos⁡(−mθ)+isin⁡(−mθ)\left( \cos \theta + i \sin \theta \right)^{-m} = \resaltar{\left( \cos(-\theta) + i \sin(-\theta) \right)^m} = \resaltar{\cos(-m\theta) + i \sin(-m\theta)}

    Si n=−mn = -m, vbi mm natvralis est, potentia exponentis −m-m est potentia mm-ta inversi, ex legibvs potentiarvm. Casvs natvralis ad angvlvm −θ-\theta adhibetvr.

Formvla ad omnes complexos non nvllos extenditvr, qvia potentia prodvcti est prodvctvm potentiarvm, etiam pro exponentibvs negativis: inversvm prodvcti est prodvctvm inversorvm.

TheoremaPotentiae in forma polari

Si z=r(cos⁡θ+isin⁡θ)z = r \left( \cos \theta + i \sin \theta \right) complexvs non nvllvs est, tvm pro omni integro nn valet zn=rn(cos⁡(nθ)+isin⁡(nθ))z^n = r^n \left( \cos(n\theta) + i \sin(n\theta) \right): modvlvs nvmeri znz^n est rnr^n, et nθn\theta est argvmentvm nvmeri znz^n. Factor realis a factore modvli 11 separatvr, et hvic formvla Moivreana adhibetvr. Figvra demonstrationem seqvitvr cvm complexo modvli pavlo maioris qvam 11 et n=3n = 3, et desinit in z=3+i=2(cos⁡π6+isin⁡π6)z = \sqrt{3} + i = 2\left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right) eivsqve tribvs primis potentiis.

Demonstratio

  1. zn=(r(cos⁡θ+isin⁡θ))n=rn(cos⁡θ+isin⁡θ)nz^n = \left( r \left( \cos \theta + i \sin \theta \right) \right)^n = \dato{p}{\resaltar{r^n} \left( \cos \theta + i \sin \theta \right)^n}

    Potentia prodvcti est prodvctvm potentiarvm: pro exponentibvs natvralibvs, ex commvtativitate et associativitate prodvcti, et pro exponentibvs negativis qvia inversvm prodvcti est prodvctvm inversorvm.

  2. zn=rn(cos⁡θ+isin⁡θ)n=rn(cos⁡(nθ)+isin⁡(nθ))z^n = r^n \resaltar{\left( \cos \theta + i \sin \theta \right)^n} = \dato{f}{r^n \resaltar{\left( \cos(n\theta) + i \sin(n\theta) \right)}}

    Secvndo factori formvla Moivreana adhibetvr, qvae pro omni integro nn valet.

  3. rn>0→∣zn∣=rn,arg⁡zn=nθ\resaltar{r^n > 0} \rightarrow \abs{z^n} = \resaltar{r^n}, \qquad \arg z^n = \resaltar{n\theta}

    Cvm r>0r > 0, potentia rnr^n est nvmervs realis positivvs, etiam pro nn negativo. Ex theoremate de aeqvalitate in forma polari, expressio obtenta est forma polaris nvmeri znz^n: eivs modvlvs est rnr^n, et nθn\theta vnvm ex eivs argvmentis.

  4. (3+i)3=23(cos⁡π2+isin⁡π2)=8i\left( \sqrt{3} + i \right)^3 = \resaltar{2^3} \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = \resaltar{8i}

    Si z=3+iz = \sqrt{3} + i, potentiae modvlos 22, 44 et 88 et argvmenta π6\frac{\pi}{6}, π3\frac{\pi}{3} et π2\frac{\pi}{2} habent: cvbvs est pvre imaginarivs.

Inde seqvitvr potentias z,z2,z3,…z, z^2, z^3, \ldots modvlos in progressione geometrica rationis rr et argvmenta in progressione arithmetica differentiae θ\theta habere, eo sensv qvo in lectione cvrsvs I de svmmationibvs et progressionibvs. Si r=1r = 1, omnes in circvlo radii 11 iacent, qvaeqve angvlo θ\theta conversa respectv eivs qvae antecedit; si r>1r > 1, convertvntvr et ab origine recedvnt; si r<1r < 1, convertvntvr et ad eam accedvnt. Praeterea, cvm zk+1=z⋅zkz^{k + 1} = z \cdot z^k et prodvctvm per zz sit conversio angvlo θ\theta cvm dilatatione rationis rr composita, omnia triangvla verticvm 00, zkz^k et zk+1z^{k + 1} similia svnt triangvlo verticvm 00, 11 et zz, vt in lectione de forma polari demonstratvm est.

ExemplvmSpira potentiarvm

Sit zz complexvs modvli 1110\frac{11}{10}, pavlo maioris qvam 11, et argvmenti π4\frac{\pi}{4}. Nonnvllas eivs potentias vsqve ad octavam compvtamvs et cvrvam describimvs in qva iacent.

Demonstratio

  1. z=1110(cos⁡π4+isin⁡π4)=11220(1+i)\dato{z}{z = \frac{11}{10} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right)} = \frac{11\sqrt{2}}{20} (1 + i)

    Forma polari et forma binomiali nvmervs sic se habet.

  2. z2=(1110)2(cos⁡π2+isin⁡π2)=121100iz^2 = \resaltar{\left( \frac{11}{10} \right)^2} \left( \cos \frac{\pi}{2} + i \sin \frac{\pi}{2} \right) = \resaltar{\frac{121}{100} i}

    Ex theoremate de potentiis in forma polari qvadratvm argvmentvm π2\frac{\pi}{2} habet: est pvre imaginarivm.

  3. z4=(1110)4(cos⁡π+isin⁡π)=−1464110000z^4 = \resaltar{\left( \frac{11}{10} \right)^4} \left( \cos \pi + i \sin \pi \right) = \resaltar{-\frac{14641}{10000}}

    Qvarta potentia argvmentvm π\pi habet: est nvmervs realis negativvs.

  4. z8=(1110)8(cos⁡2π+isin⁡2π)=214358881100000000z^8 = \resaltar{\left( \frac{11}{10} \right)^8} \left( \cos 2\pi + i \sin 2\pi \right) = \resaltar{\frac{214358881}{100000000}}

    Octava potentia argvmentvm 2π2\pi habet: post integram conversionem ad semiaxem realem positivvm redit, distantia ab origine pavlo maiore qvam 22.

  5. ∣zk+1−zk∣=∣zk(z−1)∣=∣z∣k∣z−1∣\abs{z^{k + 1} - z^k} = \resaltar{\abs{z^k \left( z - 1 \right)}} = \resaltar{\abs{z}^k \abs{z - 1}}

    Distantia inter dvas potentias consecvtivas qvoqve in progressione geometrica rationis 1110\frac{11}{10} crescit, ex theoremate modvli prodvcti. Pvncta in cvrva iacent qvae, in coordinatis polaribvs, in lectione de coordinatis polaribvs et cvrvis polaribvs tractatis, aeqvationem r=(1110)4θπr = \left( \frac{11}{10} \right)^{\frac{4\theta}{\pi}} habet: spira, logarithmica appellata, qvae, vt calcvlo differentiali ostenditvr, omnem rectam per originem dvctam semper eodem angvlo secat.