basic_tools:eulers_formula
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basic_tools:eulers_formula [2017/12/16 12:58] jakobadmin created |
basic_tools:eulers_formula [2018/03/28 13:24] jakobadmin |
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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> | ||
| Line 19: | Line 7: | ||
| </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> | ||
| Line 30: | Line 19: | ||
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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 |
| - | + | $$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = 0 -i = -i \, .$$ | |
| - | <-- | + | |
| - | <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> | ||
basic_tools/eulers_formula.txt · Last modified: 2021/05/15 18:11 by boldstonegoson
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