Let $P(z),Q(z)$ be polynomials, where $\text{deg}Q-\text{deg}P\ge 2$. Suppose $Q$ has no real roots.

Call the set of number that can be written as $p+iq$ ($p,q$ rational) as rational complex numbers.

Suppose $P,Q$ has coefficients that are rational complex numbers.

Prove or disprove:

$$\int^\infty_{-\infty}\frac{P(z)}{Q(z)}dz$$ and $\pi$ must be linearly dependent over rational complex numbers.

Complex analysis approach: Since residues of $\frac{P}Q$ are also rational complex, by residue theorem the integral and $\pi$ are linearly dependent.

Is this argument correct? If so, what is a real analysis approach to this problem?


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