Let $g(x)$ and $h(x)$ be some polynomials. What conditions ensure that the integral $$ I = \int_{-\infty}^\infty \frac{g(x)}{h(x)}dx $$ is absolutely integrable? Obviously $h(x)$ has to be of higher degree than $g(x)$, but is there a more precise condition on when this expression is absolutely integrable?
2026-05-17 09:26:05.1779009965
Integrating a ratio of polynomials $\frac{g(x)}{h(x)}$ on the real line?
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It has to be integrable at all the singularities, namely, $\infty$ and the real poles.
At infinity you have $$\frac{g(x)}{h(x)} \sim C x^{deg(g)-deg(h)}$$ So $h$ has to be of degree higer or equal to $deg(g)+2$.
If $g$ and $h$ don't have common factors, at the real poles you get $$\frac{g(x)}{h(x)} \sim \frac{C}{(x-a)^n}$$ which is never absolutly integrable. So $h$ can't have poles.