✎ Euler's Formula [The Physics Travel Guide]

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====== Euler's Formula ======

<note tip>
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.
</note>
  
<tabbox Concrete>

* For a great explanation, see [[https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/|Intuitive Understanding Of Euler’s Formula]] by Kalid Azad
  * See also [[https://betterexplained.com/articles/easy-trig-identities-with-eulers-formula/|Easy Trig Identities With Euler’s Formula]] by Kalid Azad
 
<tabbox Abstract>

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

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 [[basic_tools:exponential_function|exponential function]] and $ \cos(x)$, $\sin(x)$ are the usual [[basic_tools:trigonometric_functions|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 [[basic_tools:imaginary_numbers|imaginary unit]] $i$ and $\pi$. (Pi is the ratio between circumference and diameter shared by all circles.)

</tabbox>

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