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Exponentialis complexa et formvla Evleri

Svmmarivm

Haec lectio exponentialem nvmeri complexi definit. A dvabvs observationibvs proficiscitvr: nvmervs complexvs cos⁡θ+isin⁡θ\cos \theta + i \sin \theta svmmas angvlorvm in prodvcta convertit, sicvt potentia svmmas exponentivm in prodvcta convertit, et series exponentialis, in nvmero imaginario compvtata, in series cosinvs et sinvs dividitvr. Vtraqve ad formvlam Evleri, eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta, et ad definitionem ea+bi=ea(cos⁡b+isin⁡b)e^{a + bi} = e^a \left( \cos b + i \sin b \right) dvcit. Lectio demonstrat exponentialem complexam svmmas in prodvcta transformare, modvlvm eivs esse eae^a et argvmentvm bb, nvmqvam evanescere et periodicam esse periodo 2πi2\pi i; omnem nvmervm complexvm in forma exponentiali reiθr e^{i\theta} scribit; cosinvm et sinvm per exponentiales exprimit, ad totvm planvm extendit et in eis fvnctiones hyperbolicas detegit. Conclvditvr logarithmo complexo, qvi infinitos valores habet, et potentiis exponentis complexi, inter qvas iii^i nvmervs realis esse invenitvr.

Proposita discendi

  1. Explicare cvr lex exponentivm et series exponentialis ad formvlam Evleri dvcant, hanc compvtationem hevristicam a demonstratione distingvere et definitionem exponentialis ad omnem exponentem complexvm extendere.
  2. Proprietates exponentialis complexae demonstrare (exponentialem svmmae, modvlvm, argvmentvm et periodicitatem) et cvm forma exponentiali nvmerorvm complexorvm operari.
  3. Cosinvm et sinvm per exponentiales exprimere, ad nvmeros complexos extendere, decernere qvae identitates trigonometricae in C\mathbb{C} valere pergant, et eos cvm fvnctionibvs hyperbolicis conferre.
  4. Aeqvationem ew=ze^w = z solvere, logarithmvm principalem definire, potentias exponentis complexi compvtare et explicare cvr generatim infinitos valores habeant.

Formvla Evleri

In lectione de forma polari nvmerorvm complexorvm demonstratvm est, cvm dvo nvmeri complexi mvltiplicantvr, modvlos mvltiplicari et argvmenta addi. Pro nvmeris complexis modvli 11 theorema affirmat

(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 per E(θ)E(\theta) designatvr nvmervs complexvs cos⁡θ+isin⁡θ\cos \theta + i \sin \theta, aeqvalitas svpra scripta fit E(α)E(β)=E(α+β)E(\alpha) E(\beta) = E(\alpha + \beta), et praeterea E(0)=1E(0) = 1. Atqvi svmmas in prodvcta convertere proprietas propria potentiarvm est, ax+y=axaya^{x + y} = a^x a^y, qvae in lectione cvrsvs Algebra et Geometria I de fvnctione exponentiali tractata est; et formvla Moivreana, E(θ)n=E(nθ)E(\theta)^n = E(n\theta), in lectione de formvla Moivreana demonstrata, est lex (ax)n=anx\left( a^x \right)^n = a^{nx}. Omnia igitvr svadent E(θ)E(\theta) esse potentiam exponentis ad θ\theta proportionalis. Series exponentialis indicat qvam.

Series exponentialis cvm exponente imaginario

In lectione cvrsvs I de nvmero e demonstratvm est ee esse svmmam seriei inversorvm factorialivm. Generalivs, pro omni reali xx valent tres aeqvalitates infra scriptae, in qvibvs svmmae infinitorvm terminorvm significant, vt ibi, valorem ad qvem svmmae partiales accedvnt:

ex=∑k=0∞xkk!,cos⁡x=∑m=0∞(−1)mx2m(2m)!,sin⁡x=∑m=0∞(−1)mx2m+1(2m+1)!e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}, \qquad \cos x = \sum_{m=0}^{\infty} \frac{(-1)^m x^{2m}}{(2m)!}, \qquad \sin x = \sum_{m=0}^{\infty} \frac{(-1)^m x^{2m + 1}}{(2m + 1)!}

Hae tres series hic sine demonstratione admittvntvr: obtinentvr theoremate Tayloriano, qvod in cvrsv Calcvlvs II tractatvr, vbi praeterea ostenditvr eas convergere pro omni reali xx; convergentia primae pro omni complexo xx ad doctrinam seriervm et variabilis complexae pertinet. Series exponentialis solvm svmmas, prodvcta et divisiones adhibet, qvae in C\mathbb{C} sensvm habent, et compvtari potest in x=iθx = i\theta. Atqvi symbolvm eiθe^{i\theta} nvllam adhvc significationem habet: potentia exponentis imaginarii neqve prodvctvm factorvm aeqvalivm est neqve valor exponentialis realis. Compvtatio exempli qvod seqvitvr igitvr non est demonstratio eiθe^{i\theta} aliqvid valere, sed motivvm: ostendit svmmam seriei in iθi\theta esse cos⁡θ+isin⁡θ\cos \theta + i \sin \theta, ita vt, si series exponentialem etiam pro exponentibvs imaginariis definire debet, nvlla alia electio possibilis sit.

ExemplvmSeries exponentialis in nvmero imaginario

Sit θ\theta realis. Svbstitvimvs x=iθx = i\theta in seriem exponentialem et terminos indicis paris a terminis indicis imparis separamvs, ope cycli potentiarvm ipsivs ii ex lectione cvrsvs I de nvmeris complexis. Figvra pro θ=π\theta = \pi svmmas partiales SnS_n vt lineam fractam delineat: qvisqve terminvs est is qvi evm praecedit, qvadrante conversvs et per θk\frac{\theta}{k} mvltiplicatvs.

Solvtio

  1. ∑k=0∞(iθ)kk!=1+iθ+(iθ)22!+(iθ)33!+⋯\dato{s}{\sum_{k=0}^{\infty} \frac{(i\theta)^k}{k!}} = 1 + i\theta + \frac{(i\theta)^2}{2!} + \frac{(i\theta)^3}{3!} + \cdots

    Pro xx ponimvs iθi\theta in serie, nvllo adhvc nomine svmmae eivs dato. Dvae primae svmmae partiales svnt S0=1S_0 = 1 et S1=1+iθS_1 = 1 + i\theta.

  2. (iθ)2m=(−1)mθ2m,(iθ)2m+1=(−1)miθ2m+1\resaltar{(i\theta)^{2m}} = \dato{p}{\resaltar{(-1)^m \theta^{2m}}}, \qquad \resaltar{(i\theta)^{2m + 1}} = \dato{q}{\resaltar{(-1)^m i \theta^{2m + 1}}}

    Potentiae ipsivs ii exponentis paris svnt (−1)m(-1)^m, qvia i2m=(i2)mi^{2m} = \left( i^2 \right)^m; potentiae exponentis imparis mvltiplicando per ii obtinentvr. Termini indicis paris reales svnt, termini indicis imparis imaginarii pvri.

  3. S2N+1=∑m=0N(−1)mθ2m(2m)!+i∑m=0N(−1)mθ2m+1(2m+1)!\resaltar{S_{2N + 1}} = \dato{r}{\resaltar{\sum_{m=0}^{N} \frac{(-1)^m \theta^{2m}}{(2m)!} + i \sum_{m=0}^{N} \frac{(-1)^m \theta^{2m + 1}}{(2m + 1)!}}}

    In svmma partiali S2N+1S_{2N + 1}, qvae nvmervm finitvm terminorvm habet, terminos indicis paris seorsvm et terminos indicis imparis seorsvm colligimvs, et factorem ii ex posterioribvs extrahimvs: svmmae partiales seriervm cosinvs et sinvs prodevnt.

  4. ∑k=0∞(iθ)kk!=∑m=0∞(−1)mθ2m(2m)!+i∑m=0∞(−1)mθ2m+1(2m+1)!=cos⁡θ+isin⁡θ\sum_{k=0}^{\infty} \frac{(i\theta)^k}{k!} = \resaltar{\sum_{m=0}^{\infty} \frac{(-1)^m \theta^{2m}}{(2m)!}} + i \resaltar{\sum_{m=0}^{\infty} \frac{(-1)^m \theta^{2m + 1}}{(2m + 1)!}} = \resaltar{\cos \theta} + i \resaltar{\sin \theta}

    Dvae svmmae reales ad cosinvm et sinvm ipsivs θ\theta accedvnt, per series admissas; et S2NS_{2N} a S2N+1S_{2N + 1} termino differt qvi ad 00 accedit. Svmma seriei est nvmervs complexvs modvli 11 et argvmenti θ\theta.

  5. Sn=∑k=0n(iπ)kk!→cos⁡π+isin⁡π=−1\resaltar{S_n = \sum_{k=0}^{n} \frac{(i\pi)^k}{k!}} \rightarrow \resaltar{\cos \pi + i \sin \pi = -1}

    Pro θ=π\theta = \pi linea fracta svmmarvm partialivm in spiram vertitvr et ad −1-1 accedit: svmma S12S_{12} a −1-1 minvs qvam millesima distat.

Compvtatio svpra facta, qvae in seriebvs admissis nititvr, svmmam seriei determinat; qvod facere non potest, est demonstrare hanc svmmam esse «ee ad potentiam iθi\theta elatvm», qvia haec expressio nvllvm sensvm antea habet. Igitvr vt definitio adoptatvr.

Exponentialis imaginaria

Pro omni nvmero reali θ\theta, exponentialis imaginaria eiθe^{i\theta} est nvmervs complexvs modvli 11 et argvmenti θ\theta per formvlam Evleri datvs, qvam Leonhardvs Evlervs anno 1748 edidit.
eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos \theta + i \sin \theta

Pro θ=0\theta = 0 fit ei0=1e^{i0} = 1, vt lex exponentivm postvlat. Proprietates formae polaris et formvla Moivreana nvnc in leges potentiarvm vertvntvr, qvae iam non vt regvlae de angvlis memoria tenendae svnt.

TheoremaExponentialis imaginaria

Pro omnibvs realibvs θ\theta, α\alpha et β\beta et omni integro nn, nvmervs complexvs eiθe^{i\theta} modvlvm 11 habet; valent eiαeiβ=ei(α+β)e^{i\alpha} e^{i\beta} = e^{i(\alpha + \beta)} et (eiθ)n=einθ\left( e^{i\theta} \right)^n = e^{in\theta}; inversvm ipsivs eiθe^{i\theta} est eivs conivgatvs e−iθe^{-i\theta}, et eiα=eiβe^{i\alpha} = e^{i\beta} si et solvm si α−β\alpha - \beta est mvltiplvm integrvm ipsivs 2π2\pi. Qvaeqve affirmatio est translatio alicvivs conclvsionis lectionvm svperiorvm. Figvra pvnctvm eiθe^{i\theta} ostendit, qvod circvlvm radii 11 contra motvm horologii percvrrit dvm θ\theta crescit.

Demonstratio

  1. ∣eiθ∣=cos⁡2θ+sin⁡2θ=1=1\abs{e^{i\theta}} = \sqrt{\resaltar{\cos^2 \theta + \sin^2 \theta}} = \sqrt{\resaltar{1}} = 1

    Modvlvs est 11 per identitatem Pythagoricam lectionis de identitatibvs trigonometricis fvndamentalibvs: pvnctvm eiθe^{i\theta} est pvnctvm circvli radii 11 qvod angvlo θ\theta respondet.

  2. eiαeiβ=(cos⁡α+isin⁡α)(cos⁡β+isin⁡β)=cos⁡(α+β)+isin⁡(α+β)=ei(α+β)e^{i\alpha} e^{i\beta} = \resaltar{\left( \cos \alpha + i \sin \alpha \right) \left( \cos \beta + i \sin \beta \right)} = \resaltar{\cos(\alpha + \beta) + i \sin(\alpha + \beta)} = \dato{p}{e^{i(\alpha + \beta)}}

    Regvla prodvcti est theorema prodvcti in forma polari cvm modvlis 11 aeqvalibvs: argvmenta addvntvr.

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

    Potentia est formvla Moivreana, qvae pro omni integro nn valet.

  4. e−iθ=cos⁡θ−isin⁡θ=eiθ‾,eiθe−iθ=ei0=1e^{-i\theta} = \resaltar{\cos \theta - i \sin \theta} = \conj{e^{i\theta}}, \qquad e^{i\theta} e^{-i\theta} = \resaltar{e^{i0}} = 1

    Qvia cosinvs par est et sinvs impar, e−iθe^{-i\theta} est conivgatvs ipsivs eiθe^{i\theta}, pvnctvm ei symmetricvm respectv axis realis; et per legem prodvcti, cvm β=−α\beta = -\alpha, est etiam eivs inversvm.

  5. eiα=eiβ↔α−β=2kπ,k∈Ze^{i\alpha} = e^{i\beta} \leftrightarrow \resaltar{\alpha - \beta = 2k\pi}, \quad k \in \mathbb{Z}

    Ex aeqvalitate nvmerorvm complexorvm in forma polari, dvo nvmeri complexi modvli 11 congrvvnt si et solvm si eorvm argvmenta mvltiplo integro ipsivs 2π2\pi differvnt: post integram conversionem pvnctvm per evndem locvm itervm transit.

Identitas Evleri

Pro θ=π\theta = \pi formvla Evleri dat eiπ=cos⁡π+isin⁡π=−1e^{i\pi} = \cos \pi + i \sin \pi = -1, id est:

eiπ+1=0e^{i\pi} + 1 = 0

Haec aeqvalitas, qvae identitas Evleri appellatvr, vna relatione conivngit nvmeros 00 et 11, elementa nevtra additionis et mvltiplicationis, vnitatem imaginariam ii et dvas constantes fvndamentales analyseos, ee et π\pi. Eodem modo eiπ2=ie^{i\frac{\pi}{2}} = i, e−iπ2=−ie^{-i\frac{\pi}{2}} = -i et e2πi=1e^{2\pi i} = 1. Praesertim, qvoniam i=eiπ2i = e^{i\frac{\pi}{2}}, potentiae ipsivs ii svnt in=einπ2i^n = e^{in\frac{\pi}{2}}: cyclvs periodi 44 ex lectione cvrsvs I de nvmeris complexis est periodicitas exponentialis imaginariae, qvia qvattvor qvadrantes efficivnt 2π2\pi.