Solutions to tan(x) = x
I read something recently that said in passing that the solutions to the equation tan x = x are the zeros of the Bessel function J3/2. That brought two questions to mind. First, where have I seen the equation tan x = x before? And second, why should its solutions be the roots of a Bessel function.
The answer to the first question is that I wrote about the local maxima of the sinc function three years ago. That post shows that the derivative of the sinc function sin(x)/x is zero if and only if x is a fixed point of the tangent function.
As for why that should be connected to zeros a Bessel function, that one's pretty easy. In general, Bessel functions cannot be expressed in terms of elementary functions. But the Bessel functions whose order is an integer plus can.
For integer n,
So when n = 1, we've got the derivative of sinc right there in the definition.
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