exp(ix)=cosx+isinx really mind-blowing. If you want to integrate it you simply divide everything by i. sinx - icosx = (cosx)/i + sinx
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im just going to like this so i look more intellectual
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try typing the tweet in using eg alt+ numlock keys eg alt+ 171 ½ , alt+167 º , alt+248 etc °¨·¹³²
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I remember the night where I did the problem showing the Taylor expansion of cos and sin can be added to get exp -- blew my mind.
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can also prove with taylor expansions
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Taylor's expansions is easier.
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Or g(x)=cosX+isinX, g’(x)=i(cosX+isinX)=ig(x), g(x)=exp(ix)+const, g(0)=1, const=0, so g(x)=cosX+isinX=exp(ix).
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Leading of course to e^(iPi)=-1, but also e^(iPi/2)=i.
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