Proposition 8

Put $F(x) = \int_x^{+\infty} f(t)\,dt$ if $\int_a^{+\infty} f(t)\,dt$ converges, and $F(x) = \int_a^x f(t)\,dt$ if $\int_a^{+\infty} f(t)\,dt$ is infinite. Suppose that $\log|f|$ and $\log x$ are comparable of order 1.

1 If $f$ is of finite order $\mu \neq -1$ with respect to $x$ one has

(1)

2 If $f$ is of infinite order relative to $x$ and if $f/f'$ and $x$ are comparable of order 1, then one has

(2)

One should note that the hypothesis implies that $f(x)$ has a determinate order relative to $x$ (V, p. 219).

1 If $f$ is of order $\mu \neq 0$ relative to $x$ one has $\log|f| \sim \mu \log x$, so, since $\log|f|$ and $\log x$ are comparable of order 1, by prop. 7 of V, p. 232 one has $f'/f \sim$ μ/x, whence $x f' \sim \mu f$. If $\mu > -1$ one has $f(x) \gg x^{\mu-\varepsilon}$ for every $\varepsilon > 0$, and hence (V, p. 229, prop. 2) the integral $\int_a^{+\infty} f(t)\,dt$ is infinite. One can write $F(x) = \int_a^x f(t)\,dt = x f(x) - af(a) - \int_a^x t f'(t)\,dt$, whence again

since $\mu \neq -1$ we have $f(x) + x f'(x) \sim (\mu + 1) f(x)$, so (V, p. 230, prop. 6) which proves (1) in this case. If $\mu = 0$ one has likewise that $x f'(x) \ll f(x)$, which again gives $f(x) + x f'(x) \sim f(x)$. One argues similarly when $\mu < -1$, in the case where $\int_a^{+\infty} f(t)\,dt$ converges.

2) If $f$ is of order $+\infty$ relative to $x$ one has $\log |f| \gg \log x$, so (V, p. 232, prop. 7) $f'/f \gg 1/x$, or again, putting $g(x) = f(x)/f'(x)$, $g(x) \ll x$; further, since $f(x) \gg x^\alpha$ for every $\alpha > 0$ the integral $\int_a^{+\infty} f(t)\,dt$ is infinite. One can write

since $g$ and $x$ are comparable of order 1, from the relation $g(x) \ll x$ one can deduce (V, p. 232, prop. 7) that $g'(x) \ll 1$, hence that $f g' \ll f$, and consequently (V, p. 230, prop.6) which establishes the relation (2). The proof is similar when $f$ is of order $-\infty$ relative to $x$, in the case where $\int_a^{+\infty} f(t)\,dt$ converges.

Let $\mathscr{E}$ be a comparison scale (for real $x$ tending to $+\infty$) formed of nonzero real functions that are of constant sign on a neighbourhood of $+\infty$, such that $x \in \mathscr{E}$ and such that the product and quotient of two functions in $\mathscr{E}$ again belongs to $\mathscr{E}$ (V, p. 221 and p. 224). If a regulated function $f$ with constant sign on a neighbourhood of $+\infty$ has a principal part $c g$ relative to $\mathscr{E}$, then $\int_x^{+\infty} f(t)\,dt$ (resp. $\int_a^x f(t)\,dt$ according to the case) will be equivalent to $c \int_x^{+\infty} g(t)\,dt$ (resp. $c \int_a^x g(t)\,dt$); if the function $g$ satisfies the hypotheses of prop. 8 of V, p. 233, and if (when the formula (2) of V, p. 233, applies) one knows a principal part of $g'$ relative to $\mathscr{E}$, one will thus have a principal part of $\int_x^{+\infty} f(t)\,dt$ (resp. $\int_a^x f(t)\,dt$) relative to $\mathscr{E}$.