basic_tools:eulers_formula
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basic_tools:eulers_formula [2018/03/28 13:24] jakobadmin |
basic_tools:eulers_formula [2020/04/02 13:44] 2a02:a03f:440d:8300:a8c3:ec79:86fc:1cfb |
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| 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 | 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 | ||
| - | $$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = 0 -i = -i \, .$$ | + | $$ e^{i\pi } = \cos(\pi) + i \sin(\pi) = -1 -i 0 = -1 \, .$$ |
| 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.) | 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.) | ||
basic_tools/eulers_formula.txt · Last modified: 2021/05/15 18:11 by boldstonegoson
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