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Thread: by using ML- inequality

  1. #1
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    by using ML- inequality

    prove that :
    $\displaystyle \mid \int \frac{1}{{z^4}} dz \mid \leq 4 \sqrt 2 $ on c where C: the line segment from z=i to z=1
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  2. #2
    Super Member Random Variable's Avatar
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    $\displaystyle \Big|\int_{C} \frac{1}{z^{4}} \ dz\Big| \le ML $

    where M is an upper bound of $\displaystyle \Big|\frac{1}{z^{4}}\Big| $ on C and L is the length of C.


    the length of C is $\displaystyle \sqrt{2} $


    $\displaystyle \Big|\frac{1}{z^{4}}\Big| = \frac{1}{|z|^{4}} $

    where z is the distance from a point on C to the origin


    To maximize $\displaystyle \frac{1}{|z|^{4}} $ we have to minimize $\displaystyle |z|^{4} $

    The shortest distance from a point on C to the origin is $\displaystyle \frac{1}{\sqrt{2}} $. (It's the length of the perpendicular bisector of C.)

    so $\displaystyle |z^{4}| \ge \Big(\frac{1}{\sqrt{2}}\Big)^4 = \frac{1}{4} $

    which implies that $\displaystyle \frac{1}{|z|^{4}} \le 4 $

    then $\displaystyle \int_{C} \frac{1}{z^{4}} \ dz \le 4 \sqrt{2} $
    Last edited by Random Variable; Aug 5th 2009 at 07:11 AM.
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