Location of the extrema of the sinc function

Question

Wikipedia provides the following expression for the location of the extrema of the sinc function:

$$x_n=q-q^{-1}-\dfrac{2q^{-3}}{3}-\dfrac{13q^{-5}}{15}-\dfrac{146q^{-7}}{105}...$$

with $q =\left(n + \dfrac 12\right)\pi$

However, I have not been able to find this expression in the literature. Can anyone please provide some references where this expansion is derived, or at least mentioned?

Answer 1

Based on some proceedings from a 2006 MAA chapter meeting, the asymptotic expansion for the roots of $$\tan x=x$$ was independently produced by Euler (1748, pp 318–320), Cauchy (1827, pp 277–278 in his complete works), and Rayleigh (1877, pp 278—279).

• Cauchy, Augustin–Louis (1827). Théorie de la Propagation des Ondes à la Surface d’un Fluide Pesant d’une Profondeur Indéfinie.
• Euler, Leonhard (1748). Introductio in Analysin Infinitorum: Tomus Segundus.
• Rayleigh, John William Strutt (1877). The Theory of Sound: Volume 1. Macmillan.