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 |
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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. | ||

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<tabbox Abstract> | <tabbox Abstract> | ||

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</note> | </note> | ||

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<tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||

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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.) | ||

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+ | <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]]</cite></blockquote> | ||

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+ | <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]]</cite></blockquote> | ||

</tabbox> | </tabbox> | ||

basic_tools/eulers_formula.1618736749.txt.gz · Last modified: 2021/04/18 11:05 by cleonis

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