# Identitat d'Euler

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L'expressió

$e^{i \cdot \pi} + 1 = 0$

anomenada identitat d'Euler, és una fórmula de matemàtiques (batejada pel físic estatunidenc Richard Feynman), que uneix de forma senzilla diversos camps d'aquesta disciplina:

Aquesta identitat és un cas particular de la fórmula d'Euler:

$e^{x+i \cdot y} = e^x \cdot (\cos y + i \cdot \sin y )$

per a x = 0 i y = π. S'anomena així com a homenatge a Leonard Euler

típica imatge representativa de la Identitat d'Euler

## Explicació

Formula d'Euler general per a un angle

La identitat d'Euler és un cas especial de la Fórmula d'Euler des del punt de vista de l'Anàlisi complexa, que afirma que per qualsevol Nombre real x,

$e^{ix} = \cos x + i\sin x \,\!$

on els valors de les funcions trigonometriquessinus i cosinus venen donades en radiants.

En particular quan: x = Plantilla:Pi, o mig gir (180°), al voltant d'una circumferència :

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

Ja que

$\cos \pi = -1 \, \!$

i

$\sin \pi = 0,\,\!$

it follows that

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

which yields Euler's identity:

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

The physical explanation of Euler's identity is that it can be viewed as the group-theoretical definition of the number Plantilla:Pi. The following discussion is at the physical level, but can be made mathematically strict. The "group" is the group of rotations of a plane around 0. In fact, one can write

$e^{i \pi} = (e^{i \delta})^{\pi / \delta},\,\!$

with Plantilla:Delta being some small angle.

The last equation can be seen as the action of consecutive small shifts along a circle, caused by the application of infinitesimal rotations starting at 1 and continuing through the total length of the arc, connecting points 1 and −1 in the complex plane. Each small shift may then be written as

$1 + i \delta \!$

and the total number of shifts is Plantilla:Pi/Plantilla:Delta. In order to get from 1 to −1, the total transformation would be

$(1 + i \delta)^{\pi / \delta}. \!$

Taking the limit when Plantilla:Delta → 0, denoting iPlantilla:Delta = 1/n and the equation $e = \lim_{n\rightarrow \infty}\left(1+ {1 \over n}\right)^n$, we arrive at Euler's identity.

Plantilla:Pi is defined as the total angle which connects 1 to −1 along the arc. Therefore, the relation between Plantilla:Pi and e arises because a circle can be defined through the action of the group of shifts which preserve the distance between two points on the circle.

This simple argument is the key to understanding other relations involving Plantilla:Pi and e.

## Explicació

Formula d'Euler per a un angle general

La identitat d'Euler és un cas especial de la Formula d'Euler en l'anàlisi complexa, que afirma que per cada un dels nombres reals x,

$e^{ix} = \cos x + i\sin x \,\!$

On els valors de ttrigonometric functions sin and cosin estan donats en radiants.

In particular, when x = Plantilla:Pi, or one half-turn (180°) around a circle:

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

Since

$\cos \pi = -1 \, \!$

and

$\sin \pi = 0,\,\!$

it follows that

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

which yields Euler's identity:

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

The physical explanation of Euler's identity is that it can be viewed as the group-theoretical definition of the number Plantilla:Pi. The following discussion is at the physical level, but can be made mathematically strict. The "group" is the group of rotations of a plane around 0. In fact, one can write

$e^{i \pi} = (e^{i \delta})^{\pi / \delta},\,\!$

with Plantilla:Delta being some small angle.

The last equation can be seen as the action of consecutive small shifts along a circle, caused by the application of infinitesimal rotations starting at 1 and continuing through the total length of the arc, connecting points 1 and −1 in the complex plane. Each small shift may then be written as

$1 + i \delta \!$

and the total number of shifts is Plantilla:Pi/Plantilla:Delta. In order to get from 1 to −1, the total transformation would be

$(1 + i \delta)^{\pi / \delta}. \!$

Taking the limit when Plantilla:Delta → 0, denoting iPlantilla:Delta = 1/n and the equation $e = \lim_{n\rightarrow \infty}\left(1+ {1 \over n}\right)^n$, we arrive at Euler's identity.

Plantilla:Pi is defined as the total angle which connects 1 to −1 along the arc. Therefore, the relation between Plantilla:Pi and e arises because a circle can be defined through the action of the group of shifts which preserve the distance between two points on the circle.

This simple argument is the key to understanding other relations involving Plantilla:Pi and e.

## Explicació

Formula d'Euler per a un angle general

La identitat d'Euler és un cas especial de la Formula d'Euler en l'anàlisi complexa, que afirma que per cada un dels nombres reals x,

$e^{ix} = \cos x + i\sin x \,\!$

On els valors de ttrigonometric functions sin and cosin estan donats en radiants.

In particular, when x = Plantilla:Pi, or one half-turn (180°) around a circle:

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

Since

$\cos \pi = -1 \, \!$

and

$\sin \pi = 0,\,\!$

it follows that

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

which yields Euler's identity:

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

The physical explanation of Euler's identity is that it can be viewed as the group-theoretical definition of the number Plantilla:Pi. The following discussion is at the physical level, but can be made mathematically strict. The "group" is the group of rotations of a plane around 0. In fact, one can write

$e^{i \pi} = (e^{i \delta})^{\pi / \delta},\,\!$

with Plantilla:Delta being some small angle.

The last equation can be seen as the action of consecutive small shifts along a circle, caused by the application of infinitesimal rotations starting at 1 and continuing through the total length of the arc, connecting points 1 and −1 in the complex plane. Each small shift may then be written as

$1 + i \delta \!$

and the total number of shifts is Plantilla:Pi/Plantilla:Delta. In order to get from 1 to −1, the total transformation would be

$(1 + i \delta)^{\pi / \delta}. \!$

Taking the limit when Plantilla:Delta → 0, denoting iPlantilla:Delta = 1/n and the equation $e = \lim_{n\rightarrow \infty}\left(1+ {1 \over n}\right)^n$, we arrive at Euler's identity.

Plantilla:Pi is defined as the total angle which connects 1 to −1 along the arc. Therefore, the relation between Plantilla:Pi and e arises because a circle can be defined through the action of the group of shifts which preserve the distance between two points on the circle.

This simple argument is the key to understanding other relations involving Plantilla:Pi and e.