Example 2

Tag 08H7, permanent. Cite it as [Bourbaki, Tag 08H7].

Functions of a Real Variable, chapter V, Appendix 0. Hardy Fields. (H) Functions

Let us seek an asymptotic expansion for the inverse function of the function $x / \log x$ ; from the identity $x = y / \log y$ , where $y = v(x)$ , one deduces $\log x = \log y - \log \log y$ ; on putting $z = \log y, t = \log x$ , one has $t = z - \log z$ , and one is led to expand the inverse function $u(t)$ of $t - \log t$ ; for example, one has

u_2(t) = t + \log(t + \log t) = t + \log t + \frac{\log t}{t} - \frac{(\log t)^2}{2t^2} + o_1\left( \frac{\log t}{t^2} \right)

and moreover, by (11) (V, p. 256)

u(t) - u_2(t) \sim \frac{\log t}{t^2}

whence

u(t) = t + \log t + \frac{\log t}{t} - \frac{(\log t)^2}{2t^2} + \frac{\log t}{t^2} + o_2\left( \frac{\log t}{t^2} \right)

and on returning to the original problem one obtains the asymptotic expansion

v(x) = x \log x + x \log \log x + x \frac{\log \log x}{\log x} + o\left( x \frac{\log \log x}{\log x} \right).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

V, p. 256