Euleri identitas

In complexorum numerorum analysi, Euleri identitas

${\displaystyle e^{i\pi }=-1\,\!}$

aequatio est, in quo

${\displaystyle e}$ Numerus Euleri, logarithmi naturalis basis, est

${\displaystyle i}$ Unus Imaginarius, complexus numerus cuius radix -1, est

${\displaystyle \pi }$ Pi Graecus, circuli diametri cum eius circumscriptione ratio, est

Aliquando Euleri identitas scribitur etiam:

${\displaystyle e^{i\pi }+1=0\,\!}$

ut relationem ex his quinque praecipuis mathematicis constantibus elementis illustret.

In Introductione, in Lausanna anno 1748 divulgata, Leonhardus Eulerus hanc aequationem scripsit. Euleri identitas generalis Euleri formulae singularis casus est

${\displaystyle e^{ix}=\cos x+i\sin x\qquad \forall x\in \Re \!}$

Si ${\displaystyle x=\pi }$, tum

${\displaystyle e^{i\pi }=\cos \pi +i\sin \pi \,\!}$

inde, ex definitione

${\displaystyle \cos \pi =-1\,}$

et

${\displaystyle \sin \pi =0\,}$

ergo

${\displaystyle e^{i\pi }=-1\,\!}$

Beniaminus Peirce, undevicesimo saeculo mathematicus et in Harvardi doctor, postquam in schola identitatem argumentis confirmavit, inquit: "Vos qui audite, haec identitas certe vera, sed maxime admiramibilis est; hanc comprehendere non possumus, nec quid significet scimus. At hanc demonstravimus, et ob eam rem hanc veritatem esse debere scimus."

Richardus Feynman Euleri formulam (ex quo Euleri identitas deducitur) "maximam formulam in mathematica" definivit. Feynman plurimique alii mathematici hanc identitatem notabilem putant quia aliquae mathematica constantia elementa iungit:

• Numerus 0, per summam identitatis elementum (${\displaystyle \forall a,\,a+0=0+a=a}$).
• Numerus 1, per multiplicationem identitatis elementum (${\displaystyle \forall a,\,a\cdot 1=1\cdot a=a}$).
• Numerus π trigonometriae, Euclidis geometriae analyseosque praecipuum constans elementum.
• Numerus e logarithmorum studiorum analyseosque praecipuum constans elementum (e.g.: in incrementi functionum descriptione vel in discriminum simplicis aequationis explicatione).
• Numerus Unus Imaginarius i (i² = −1), complexorum numerorum numerus. Unus Imaginarius facultatem nobis solvendi omnes polynomiales aequationes in complexorum numerorum regione dat.

Postremo, omnes praecipui arithmetici signi sunt: aequalitas, summa, multiplicatio, elatio.

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