it is genuinely astonishing how often this works. physicists will literally perform taylor series expansions in a parameter that is like ~1 and somehow this doesn't produce total garbage. completely mysterious to me
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the most based calculation i've ever seen in physics is a statmech calculation that works in "4 - epsilon" dimensions where epsilon is "assumed to be small" (for taylor expansions etc) and then set equal to 1, as a way to guess what happens in 3 dimensions. incredible
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details available e.g. here, "the epsilon expansion":
damtp.cam.ac.uk/user/tong/sft.
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the best bit about my engineering degree is it gives me a license to abuse mathematics, for a living
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alas that doesn't pay as well as abusing computers instead so here i am
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That one shows up in quantum field theory too.
The worst part is that other math that isn’t as fucked up agrees with it
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my advisor told me about this last week, honestly the epsilon part isn't what got me (as far as i can tell you're just making an integer-valued variable real-valued, it's not that weird). the weird part is that the *dimensional analysis gives the wrong answer*
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i'm still fuckin strapped over that, he didn't explain the whole thing and i don't have time to dive into a research hole right now so i don't even understand how that's possible
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Typically most of the series in physics are asymptotic in the sense of Poincare. Truncation of such series up to or below an optimal point works well.
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Nicholas de Bruijn has a book which purports to explain how this stuff actually works. (I'm sure you could get through it faster than I can.)