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basic_tools:eulers_formula

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# Euler's Formula

## Why is it interesting?

Euler's formula is a deep result of complex analysis that shows a surprising relationship between seemingly unrelated notions:

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

where $e^{ix}$ denotes the exponential function and $\cos(x)$, $\sin(x)$ are the usual trigonometric functions. If we evaluate this equation at $x= \pi$, we get

$$e^{i\pi } = \cos(\pi) + i \sin(\pi) = 0 -i = -i \, .$$

This shows a deep relationship between the exponential function, the imaginary unit $i$ and $\pi$. (Pi is the ratio between circumference and diameter shared by all circles.)

## Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Researcher

The motto in this section is: the higher the level of abstraction, the better.

Example1
Example2: