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 ½ can.

For integer n,

J_{n+{frac{1}{2}}}(x)= (-1)^n left(frac{2}{pi}right)^{frac{1}{2}} x^{n+{frac{1}{2}}} left(frac{1}{x}frac{d}{dx}right)^nleft(frac{sin x}{x}right)

So when n = 1, we’ve got the derivative of sinc right there in the definition.

The post Solutions to tan(x) = x first appeared on John D. Cook.

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