How do I understand Euler's identity

Euler's formula / Euler's identity

In 1748 Leonhard Euler proved in the context of his work Introductio in analysin infinitorum the so-called Euler's identity. The following equation applies to real numbers x:

Euler's formula connects the natural exponential function e in the complex number spacex with the trigonometric functions sin (x) and cos (x). First of all, that's pretty amazing and anything but trivial. The connection can be seen quite well with the help of the Tayler series development of the respective functions.

The power series of cosine (i * x) and sine (i * x) are very similar to that of ei x.

Who looks closely and i2 = -1 is taken into account, who will recognize that the components of the sine and cosine series can be added up exactly to the components of the E-function series. The derivation via the power series also forms the basis for the proof of the Euler formula.

A special case of Euler's formula or identity is the case x = π. If we insert the circle number pi into Euler's equation, we get

ei * π = -1

If that's not really amazing ...

Isn't that nice? It becomes even more beautiful after a slight reshaping: eπi + 1 = 0. This application of Euler's formula combines five of the most important numbers in mathematics in one formula. This particular equation is often referred to as Euler's identity.