Random question: does there exist an analytical expression for the first positive zero of J₀(x) ~ 2.4048255577?
More or less: h(x) sproings the real line so all the roots are in the appropriate places, and u(x) scales it so that its.extrema are +-1 J(0,h(x)) * u(x) = cos(x)
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Yeah, I'm familiar with modulus-phase decompositions for Bessel/Airy functions, SLATEC uses them for x > ~8 & I use them for Airy f'ns myself, it's just this *does* reduce the problem to finding an analytic expression for the phase function h(x). We *have* asymptotics for it…
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…(fairly vile ones, honestly, the coefficients rapidly become an utter mess), but nothing that seems sensible when x is small. Chebyshev/Minimax series obviously work numerically, but still feel unsatisfying in a way here. I guess I'm cosplaying a pure mathematician today.
End of conversation
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