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The identity of Euler is the following relation:

$e^\; \{I\; \backslash \; pi\}\; +\; 1\; =\; 0\; \backslash ;$

where $\backslash \; e$ is the base of the Napierian logarithm , $\backslash \; i$ is the unit of the pure imaginary (checking $\backslash \; i^2=-1$) and $\backslash \; \backslash \; pi$ is the constant of Archimedes (the report/ratio of the circumference of a Cercle with its Diamètre).

The identity appears in the book Introductio of Leonhard Euler, published with Lausanne in 1748.

In the foreword of the one of its books, whereas it was almost fifteen years old, Richard Feynman, qualified this identity of “the most remarkable formula in the world”.

Feynman found this formula remarkable because it binds constant fundamental mathematics:

• the numbers $\backslash \; 0$ and $\backslash \; 1$ are respectively the neutral elements for the Addition and the Multiplication.
• the number $\backslash \; \backslash \; pi$ is a constant relative to our Euclidean world, at least on small scales (if not the report/ratio length of the circumference of the circle to its diameter is not a universal constant, i.e. the same one for all the circumferences).
• the number $\backslash \; e$ is important in the description of the behaviors of strong growth, and appears in the solution $\backslash \; y$ ($\backslash \; there\; (X)\; =\; e^x$) of simplest the differential equation of growth: $\backslash \; Dy/dx\; =\; y$ and $\backslash \; there\; (0)\; =1$.
• Lastly, the imaginary number $\backslash \; i$ was introduced so that all the nonconstant Polynôme S with real coefficients admit roots (see the Théorème of Alembert).

The formula also comprises the fundamental arithmetic operations of addition, multiplication and rise to a power. This formula is a particular case of the Formule of Euler in Analyze complexes:

For all Real number $\backslash \; x$,

$\backslash \; e^\; \{ix\}\; =\; \backslash \; cos\; X\; +\; I\; \backslash \; sin\; X\; \backslash ,\; \backslash !$

(average mnemotechnics: cis (X) = C bone (X) + I S in (X))

If we pose $\backslash \; X\; =\; \backslash \; pi$, then

$\backslash \; e^\; \{I\; \backslash \; pi\}\; =\; \backslash \; cos\; \backslash \; pi\; +\; I\; \backslash \; sin\; \backslash \; pi\; \backslash ,\; \backslash !$

and since $\backslash \; \backslash \; cos\; (\backslash \; pi)\; =\; -1$ and $\backslash \; \backslash \; sin\; (\backslash \; pi)\; =\; 0$, we obtain

$\backslash \; e^\; \{I\; \backslash \; pi\}\; =\; -1\; \backslash ,\; \backslash !$

and consequently,

$\backslash \; e^\; \{I\; \backslash \; pi\}\; +\; 1\; =\; 0\; \backslash ,\; \backslash !$

• geometrical interpretation is resulting from $e^\; \{I\; \backslash \; pi\}\; \backslash \; simeq\; (1\; +\; \backslash \; frac\; \{I\; \backslash \; pi\}\; \{NR\})\; ^N\; \backslash \; simeq\; -1\; \backslash ;$

starting from the following germ reiterated NR time

Indeed, on the one hand, $z\; \backslash ,\; \backslash \; in\; \backslash \; mathbb\; \{C\}\; \backslash \; quad\; e^z\; =\; \backslash \; lim\_\; \{N\; \backslash \; to\; \backslash \; infty\}\; \backslash \; left\; (1+\; \backslash \; frac\; \{Z\}\; \{N\}\; \backslash \; right)\; ^n$ and on the other hand the complex multiplications resulting in rotations, the point of coordinates $\backslash \; left\; (1\; +\; \backslash \; frac\; \{I\; \backslash \; pi\}\; \{NR\}\; \backslash \; right)\; ^N$ is obtained by juxtaposing $N\; \backslash ,$ right-angled triangles as indicated on the figure opposite.

Also beautiful and mysterious that is this identity of Euler, one better geometrically includes/understands why, when $N\; \backslash ,$ tend towards $\backslash \; infty\; \backslash ,$, the point of affix $e^\; \{I\; \backslash \; pi\}\; \backslash ,$ is equal to $(-\; 1,0)\; \backslash ,$

Another identity of Euler in analysis to several variables

The identity of Euler is the following relation: (see Theorem of Euler (functions of several variables))

If $f\; (x\_1,\; x\_2,\dots ,\; x\_n)$ is a function of homogeneous class C ¹ of degree K, then

$\backslash \; sum\_\; \{i=1\}\; ^\; \{N\}\; x\_i\; \backslash \; frac\; \{\backslash \; partial\; F\}\; \{\backslash \; partial\; \{x\_i\}\}\; =k\; f$

External bond

Proof of formula of Euler with Taylor series

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