Remark

Prop. 8 does not apply directly to a function $f$ of order $-1$ relative to $x$. But then one can write $f(x) = f_1(x)/x$, with $f_1$ of order 0 relative to $x$. Suppose, for example, that $\int_a^{+\infty} f(t)\,dt$ is infinite; then

If the function $f_1(e^t)$ satisfies the hypotheses of prop. 8 and has an order $\neq -1$ relative to $y$ (that is, if $f_1(x)$ has an order $\neq -1$ relative to $\log x$) the formulae (1) and (2) again allow us to obtain a principal part for $F(x)$. For example, let $f(x) = \frac{\exp(\sqrt{\log x})}{x \log x}$; since $\exp(\sqrt{\log x})$ is of order 0 relative to $x$, $f$ is of order $-1$; here one has $f_1(e^t) = e^{\sqrt{t}}/y$, and this function is of order $+\infty$ relative to $y$; prop. 8 applies and gives $\int_a^x e^{\sqrt{u}}/u\,du \sim 2e^{\sqrt{y}}/\sqrt{y}$: on reverting to the variable $x$ we have $\int_a^x \frac{\exp(\sqrt{\log t})}{t \log t}\,dt \sim \frac{2\exp(\sqrt{\log x})}{\sqrt{\log x}}$.