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

Student

• For a great explanation, see Intuitive Understanding Of Euler’s Formula by Kalid Azad

Researcher

Examples

Example1

Example2:

FAQ

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