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basic_tools:eulers_formula

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 basic_tools:eulers_formula [2018/04/16 07:50]jakobadmin ↷ Page moved from basic_notions:eulers_formula to basic_tools:eulers_formula basic_tools:eulers_formula [2020/04/02 13:44] (current)2a02:a03f:440d:8300:a8c3:ec79:86fc:1cfb Both sides previous revision Previous revision 2020/04/02 13:44 2018/04/16 07:50 jakobadmin ↷ Page moved from basic_notions:eulers_formula to basic_tools:eulers_formula2018/03/28 13:24 jakobadmin 2017/12/16 13:25 jakobadmin [Student] 2017/12/16 12:58 jakobadmin created Next revision Previous revision 2020/04/02 13:44 2018/04/16 07:50 jakobadmin ↷ Page moved from basic_notions:eulers_formula to basic_tools:eulers_formula2018/03/28 13:24 jakobadmin 2017/12/16 13:25 jakobadmin [Student] 2017/12/16 12:58 jakobadmin created Line 27: Line 27: 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.)