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Math Help - An application of the residue theorem

  1. #1
    Aki
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    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
    <br />
\int_{-\infty}^\infty \frac{P(z)}{Q(z)} dz,<br />
    where P(z) and Q(z) are polynomials of degree m and n, respectively,
    exists if and only if
    <br />
n-m \geq 2<br />
    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.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Aki View Post
    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
    <br />
\int_{-\infty}^\infty \frac{P(z)}{Q(z)} dz,<br />
    where P(z) and Q(z) are polynomials of degree m and n, respectively,
    exists if and only if
    <br />
n-m \geq 2<br />
    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 P(x)\sim Ax^m and Q(x)\sim Bx^n when x\to\pm\infty, hence \frac{P(x)}{Q(x)}\sim\frac{A}{B}\frac{1}{x^{n-m}} when x\to\pm\infty.

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

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