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basic_tools:eulers_formula

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 basic_tools:eulers_formula [2021/04/18 11:05]cleonis Added demonstration that Euler's formula follows from the Taylor series expansions basic_tools:eulers_formula [2021/05/15 18:11] (current)boldstonegoson Both sides previous revision Previous revision 2021/05/15 18:11 boldstonegoson 2021/05/15 18:09 boldstonegoson 2021/05/15 18:04 boldstonegoson [Beauty] 2021/05/15 18:01 boldstonegoson 2021/04/18 11:05 cleonis Added demonstration that Euler's formula follows from the Taylor series expansions2020/04/02 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 2021/05/15 18:11 boldstonegoson 2021/05/15 18:09 boldstonegoson 2021/05/15 18:04 boldstonegoson [Beauty] 2021/05/15 18:01 boldstonegoson 2021/04/18 11:05 cleonis Added demonstration that Euler's formula follows from the Taylor series expansions2020/04/02 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 77: Line 77: The connection between the exponential function and the trigonometric functions is the property of getting the same function back after taking the derivative. The difference is whether the original function comes back right away, or in a 2-cycle pattern, or in a 4-cycle pattern. The connection between the exponential function and the trigonometric functions is the property of getting the same function back after taking the derivative. The difference is whether the original function comes back right away, or in a 2-cycle pattern, or in a 4-cycle pattern. - - - - - - - - - - - - - - - - - - - - - - - - - -  ​  ​ Line 110: Line 84: ​ - ​  ​  ​ Line 122: Line 95: 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.) + + <​blockquote>​ Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler'​s equation reaches down into the very depths of existence. <​cite>​[[https://​books.google.com/​books?​id=GvSg5HQ7WPcC&​pg=PA1#​v=onepage&​q&​f=false|Keith Devlin]]​ + + <​blockquote>​ [Euler'​s equation] is absolutely paradoxical;​ we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth"​. <​cite>​[[https://​books.google.com/​books?​id=eIsyLD_bDKkC&​pg=PA160|Benjamin Peirce]]