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The following results were derived from my teacher. They are useful results to evaluate any improper integral. I want to know what you think about them and answer my inquiries

Summary

(1) $\deg(q(x))>\deg(p(x))+1$, $q(x)$ does not vanish along real axis

$$\int_{-\infty}^\infty \frac{p(x)}{q(x)} \, dx=2\pi i\left(\sum_\text{upper half-plane} \operatorname{Res} \frac{p(z)}{q(z)}\right)$$

2) $\deg (q(x))>\deg(p(x))$ $a>0$ and $q(x)$ does not vanish along real axis

$$\int_{-\infty}^\infty \frac{p(x)}{q(x)}e^{iax} \, dx=2 \pi i\left(\sum_\text{upper half-plane} \operatorname{Res} \frac{p(z)}{q(z)}e^{iaz} \right)$$

3) $q(x)$ vanishes at $x=0$ with residue $B$ $$\int_{-\infty}^\infty \frac{p(x)}{q(x)}e^{iax}dx=2 \pi i \left(\sum_\text{upper half-plane} \operatorname{Res}\frac{p(z)}{q(z)}e^{iaz} \right)+\pi i B$$

what does vanish mean?. I don't want you to derive these expressions, but explain how they are true and applicable.

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Does not vanish along $\mathbb{R}$: this means for all $x \in \mathbb{R}$, $q(x) \neq 0$. Essentially, they're true by the residue theorem https://en.wikipedia.org/wiki/Residue_theorem. They're useful as you have explicit formulas to compute integrals that may not be obvious.

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