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basic_tools:eulers_formula [2017/12/16 12:58] jakobadmin created |
basic_tools:eulers_formula [2020/04/02 13:44] 2a02:a03f:440d:8300:a8c3:ec79:86fc:1cfb |
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====== Euler's Formula ====== | ====== Euler's Formula ====== | ||
- | <tabbox Why is it interesting?> | + | <tabbox Intuitive> |
- | + | ||
- | 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 Layman> | + | |
<note tip> | <note tip> | ||
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</note> | </note> | ||
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- | <tabbox Student> | + | <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 | * 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 Researcher> | + | <tabbox Abstract> |
<note tip> | <note tip> | ||
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- | <tabbox Examples> | + | <tabbox Why is it interesting?> |
- | --> Example1# | + | Euler's formula is a deep result of complex analysis that shows a surprising relationship between seemingly unrelated notions: |
- | + | $$ e^{ix} = \cos(x) + i \sin(x) \, $$ | |
- | <-- | + | |
- | --> Example2:# | + | 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 |
- | + | $$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = -1 -i 0 = -1 \, .$$ | |
- | <-- | + | |
- | <tabbox FAQ> | + | 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 History> | + | |
</tabbox> | </tabbox> | ||