Example 1

Tag 08H6, permanent. Cite it as [Bourbaki, Tag 08H6].

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

Let us seek an asymptotic expansion for the inverse function $v(x)$ of $x^5 + x$ (for $x$ tending to $+\infty$ ); putting $x^5 = t$ one is led to seek an expansion of the inverse function $u(t)$ of $t + t^{1/5}$ (for $t$ tending to $+\infty$ ), that is, to apply prop. 5 of V, p. 256, to the case where $g(t) = -t^{1/5}$ . Let us, for example, calculate, $u_2(t)$ ; one has

u_2(t) = t - \left( t - t^{1/5} \right)^{1/5} = t - t^{1/5} + \frac{1}{5} t^{-3/5} + \frac{2}{25} t^{-7/5} + o_1\left( t^{-7/5} \right).

Moreover, by (11) (V, p. 256)

u(t) - u_2(t) \sim -\frac{1}{25} t^{-7/5}

whence

u(t) = t - t^{1/5} + \frac{1}{5} t^{-3/5} + \frac{1}{25} t^{-7/5} + o_2\left( t^{-7/5} \right)

and one deduces the expansion

v(x) = u(x)^{1/5} = x^{1/5} - \frac{1}{5} x^{-3/5} - \frac{1}{25} x^{-7/5} + o_3\left( x^{-7/5} \right)

sought.

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