Math people. Do you have an intuitive *geometric* reason why interlacing the power series of sin x and cos x gives the series for eΛ£?
Do you have a way that you *visualize* growth whose rate is exactly its present value coming out of the unit circle as sine and cosine do?
I know the equations, and the proofs and they aren't convoluted and probably "enough" but I hope someone can share their mental images. Even if they aren't fully formed.
https://sauropods.win/@futurebird/113532721775708525
@futurebird Well, the first point is that the function exp(ix) has an βorthogonalβ growth rate, since it satisfies the equation
exp(ix)' = i exp(ix)
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2024-11-24 johncarlosbaez β 1π
@futurebird - I visualize this stuff as part of the approach to trig described here:
math.ucr.edu/home/baez/trig.htβ¦
@futurebird I've watched a bunch of YouTube videos on the subject now. My conclusion is that youtu.be/B1J6Ou4q8vE is by far the most action packed discussion of this. Also if I understand you [β¦]
@futurebird I've just internalized that e^ix is a helix, sin and cos are its shadows on coordinate planes
@futurebird Oddly, because I do rely on visualization in a lot of situations in the cases you ask about I donβt, but I can intuit the answer - in the sense I can give a decent guess and look at [β¦]
@futurebird
I never got deep into power series so I can't offer any insights on that side of things, but intuitively, "growth" is just another name for the slope, plus "slope is rise over run" [β¦]
2024-11-23 adredish β 1π 1π€
@futurebird
@albertcardona
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2024-11-23 Zamfr β edited β 2π 1π¬
@futurebird
There is a nice attempt in Visual complex analysis, by Tristan Needham.
He first encourages you to see exp(x) as (1 + x/n)**n for n-> infinity. Interest paid out continuously, [β¦]
@futurebird not sure this goes anywhere yet but I suspect the fact they are orthogonal will play into it....
@futurebird RIP Ralph Abraham who could draw a single picture to explain complex ideas in analysis and algebra that would have certainly explained this. He was an amazing teacher.
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