I'm really not a π day hater, but I do roll my eyes when I read articles exclaiming π "goes on forever!" as if that is a rare and notable property that makes it unique among numbers. I mean, 1/3 "goes on forever" too.
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Replying to @divbyzero
What's really needed is a better appreciation of Gotthold Eisenstein's penetrating analysis of functions of period 1. When everyone knows the wonders lurking in ... + 1/(x - 2) +1/(x - 1) + 1/x + 1/(x + 1) + 1/(x + 2) + ... then we will indeed all have cause to celebrate.
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Replying to @MathPrinceps @divbyzero
That's pi*cot(pi*x), you can prove it using complex analysis
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Some Googling says this was first discovered by Euler (presumably using methods other than complex analysis)
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Replying to @akivaw @divbyzero
Eisenstein's methods reach extremely far, and indeed inaugurated a profound sequence of developments in analytic number theory that continue to this day. Yet they relied essentially on "only high school algebra." It's not the tool that matters; it's the touch of a master's hand.
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The really crucial point, though, which I sought tacitly to make earlier, is that "pi" should be defined as the constant appearing in "pi*cot(pi*x)." This is the "morally right" place to root our understanding of the number that so many people seem inclined to celebrate today.
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