快速了解欧拉公式

Euler’s Formula

There’re totally two kinds of Euler’s formula:

The 1st one:

a topological invariance——F + V = E + 2, where F is the number of faces, V the number of vertices, and E the number of edges.

A cube, for example, has 6 faces, 8 vertices, and 12 edges, and satisfies this formula.

The 2nd one:

 

Figure1

The formula is still valid if x is a complex number.

The original proof is based on the Taylor series expansions of e^x and of sin x and cos x for real numbers x:

 


The three equations above yields what was shown in figure1.

Euler's formula provides a means of conversion between cartesian coordinates and polar coordinates.

PS: Complex plane can be considered as a modified Cartesian plane.


Preferences:

https://en.wikipedia.org/wiki/Euler%27s_formula

https://www.britannica.com/science/Eulers-formula

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