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basic_tools:eulers_formula [2017/12/16 13:25]
jakobadmin [Student]
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>
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 </​note>​ </​note>​
   ​   ​
-<​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   * 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
  
-  +$$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = 0 -i = -i  \,  .$$
-<--+
  
-<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