Library / Functions of a Real Variable / Chapter V / § 2
Example
Tag 08F2, permanent. Cite it as [Bourbaki, Tag 08F2].
Functions of a Real Variable, chapter V, § 2. ASYMPTOTIC EXPANSIONS
We have x 1 / 1 = exp ( log x ⋅ exp ( 1 x log x ) ) x^{1/1} = \exp \left( \log x \cdot \exp \left( \frac{1}{x} \log x \right) \right) x 1/1 = exp ( log x ⋅ exp ( x 1 log x ) ) x x x + ∞ +\infty + ∞ log x ≪ x \log x \ll x log x ≪ x log x ⋅ exp ( 1 x log x ) \log x \cdot \exp \left( \frac{1}{x} \log x \right) log x ⋅ exp ( x 1 log x ) x α ( log x ) β x^\alpha (\log x)^\beta x α ( log x ) β
log x ⋅ exp ( 1 x log x ) = log x + ( log x ) 2 x + 1 2 ( log x ) 3 x 2 + o ( ( log x ) 3 x 2 ) .
\log x \cdot \exp \left( \frac{1}{x} \log x \right) = \log x + \frac{(\log x)^2}{x} + \frac{1}{2} \frac{(\log x)^3}{x^2} + o \left( \frac{(\log x)^3}{x^2} \right).
log x ⋅ exp ( x 1 log x ) = log x + x ( log x ) 2 + 2 1 x 2 ( log x ) 3 + o ( x 2 ( log x ) 3 ) . All the terms of this expansion, starting from the second, tend to 0; from this expansion and from the Taylor expansion e u = 1 + u + u 2 / 2 + o ( u 2 ) e^u = 1 + u + u^2/2 + o(u^2) e u = 1 + u + u 2 /2 + o ( u 2 ) u = 0 u = 0 u = 0
x 1 / 1 = 1 + ( log x ) 2 + 1 2 ( log x ) 4 x + 1 2 ( log x ) 3 x + o ( ( log x ) 3 x ) .
x^{1/1} = 1 + (\log x)^2 + \frac{1}{2} \frac{(\log x)^4}{x} + \frac{1}{2} \frac{(\log x)^3}{x} + o \left( \frac{(\log x)^3}{x} \right).
x 1/1 = 1 + ( log x ) 2 + 2 1 x ( log x ) 4 + 2 1 x ( log x ) 3 + o ( x ( log x ) 3 ) . Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Transcribed from the printed volume. Not translated.