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Weierstrass Factorization Theorem: Every entire function 𝑓(𝑧) can be 'factored' into a (possibly infinite) product of polynomials with the same zeroes as 𝑓(𝑧)pic.twitter.com/Te2YUrgP67
This is how Euler 'proved' his solution to the famous Basel problem I say 'proved' because Weierstrass Factorization Theorem wouldn't be proven until ~100 years laterhttps://twitter.com/InertialObservr/status/1082487078913433600?s=20 …
Not exactly true. Only holds true for order <1 entire functions, otherwise you need exponential factors. In the Euler examples you can pair opposite root factors, or equivalently sin(sqrt(z))/sqrt(z) is of order 1/2<1.
But these expansions do include the exponential factors. I just happened to pick a couple examples where they simplify nicelypic.twitter.com/gJMH3QGlHr
Interesting! What does it mean for a function to be entire?
Means analytic over all C
Witchcraft
I have a derivation giving an expression for the case n^3, but the margin of this tweet is too small to contain it ...
This one is kinda fuckin me up rn ngl
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