Approximating the locations of the extrema of the sinc function

Question

The wikipedia article on the sinc function gives an approximation for the (positive $x$) locations of the extrema of the sinc function as

$x_n \simeq (n + \frac{1}{2})\pi - \frac{1}{(n + \frac{1}{2})\pi}$

but provides no reference or proof. How would I go about deriving such an approximation?

Answer 1

The derivative is $\frac{\cos{x}}{x}-\frac{\sin{x}}{x^2}$, so you want to find solutions to $x=\tan{x}$. Draw a picture to see what's going on, then expand $\tan{x}$ about $x=(n+1/2)\pi$ to get the next term.