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Math Help - by using ML- inequality

  1. #1
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    by using ML- inequality

    prove that :
     \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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     \Big|\int_{C} \frac{1}{z^{4}} \ dz\Big| \le ML

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


    the length of C is  \sqrt{2}


     \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  \frac{1}{|z|^{4}} we have to minimize  |z|^{4}

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

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

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

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