# Thread: An application of the residue theorem

1. ## An application of the residue theorem

I have a question about an application of the residue theorem to real definite integrals.
In "COMPLEX ANALYSIS" by Ahlfors, there is the following statement:

The integral
$\displaystyle \int_{-\infty}^\infty \frac{P(z)}{Q(z)} dz,$
where P(z) and Q(z) are polynomials of degree m and n, respectively,
exists if and only if
$\displaystyle n-m \geq 2$
and Q(z) has no zeros on real axis.

A brief proof of the sufficiency follows the statement.
But, there isn't the proof of the necessity.
And, I couldn't find it in any other book on complex analysis.
I'm grateful if someone could give me a proof of the necessity of the statement above.

2. Originally Posted by Aki
I have a question about an application of the residue theorem to real definite integrals.
In "COMPLEX ANALYSIS" by Ahlfors, there is the following statement:

The integral
$\displaystyle \int_{-\infty}^\infty \frac{P(z)}{Q(z)} dz,$
where P(z) and Q(z) are polynomials of degree m and n, respectively,
exists if and only if
$\displaystyle n-m \geq 2$
and Q(z) has no zeros on real axis.

A brief proof of the sufficiency follows the statement.
But, there isn't the proof of the necessity.
And, I couldn't find it in any other book on complex analysis.
I'm grateful if someone could give me a proof of the necessity of the statement above.
You don't need the residue theorem for this. It is actually very simple: we have $\displaystyle P(x)\sim Ax^m$ and $\displaystyle Q(x)\sim Bx^n$ when $\displaystyle x\to\pm\infty$, hence $\displaystyle \frac{P(x)}{Q(x)}\sim\frac{A}{B}\frac{1}{x^{n-m}}$ when $\displaystyle x\to\pm\infty$.

On the other hand, $\displaystyle \int_1^\infty\frac{dx}{x^\alpha}$ converges if and only if $\displaystyle \alpha>1$.

By the comparison property, we deduce that $\displaystyle \int_{-\infty}^{+\infty}\frac{P(x)}{Q(x)}dx$ converges if and only if $\displaystyle n-m>1$. Since $\displaystyle n-m$ is an integer, the condition is equivalent to $\displaystyle n-m\geq 2$.