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Math Help - Question on Rouche Theorem -1

  1. #1
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    Question on Rouche Theorem -1

    f(z) = z^5 + 15z +1

    Prove all roots of f(z) are inside |z| <1

    Question says use Rouche Theorem

    Can someone help me get started on this?
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  2. #2
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    Quote Originally Posted by aman_cc View Post
    f(z) = z^5 + 15z +1

    Prove all roots of f(z) are inside |z| <1

    Question says use Rouche Theorem

    Can someone help me get started on this?
    see this link Rouché's theorem - Wikipedia, the free encyclopedia


    First by the fundemental theorem of Algebra f has 5 zero's including multiplicity and Consider the Unit Disk.

    Define g(z)=z^5 and note that g(z) has 5 zero's in the unit disk (i.e 0 is a root of multiplicity 5)

    Now lets check Rouche's theorem |f(z)-g(z)|< |f(z)|+|g(z)|

    |f(z)-g(z)|=|15z+1| \le 15|z|+1 \le 16 on the boundry of the unit disk

    and |f(z)|+|g(z)|=|z^5+15z+1|+|z^5|\ge 2|z^5|+15|z|+1 \ge 18

    so then

    |f(z)-g(z)|< |f(z)|+|g(z)| on the boundry so they gave the same number of zero's on the unit disk.

    So f has 5 zero's inside the unit disk. Yay
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    see this link Rouché's theorem - Wikipedia, the free encyclopedia


    First by the fundemental theorem of Algebra f has 5 zero's including multiplicity and Consider the Unit Disk.

    Define g(z)=z^5 and note that g(z) has 5 zero's in the unit disk (i.e 0 is a root of multiplicity 5)

    Now lets check Rouche's theorem |f(z)-g(z)|< |f(z)|+|g(z)|

    |f(z)-g(z)|=|15z+1| \le 15|z|+1 \le 16 on the boundry of the unit disk

    and |f(z)|+|g(z)|=|z^5+15z+1|+|z^5|\ge 2|z^5|+15|z|+1 \ge 18

    so then

    |f(z)-g(z)|< |f(z)|+|g(z)| on the boundry so they gave the same number of zero's on the unit disk.

    So f has 5 zero's inside the unit disk. Yay


    If we put z = -1 aren't we getting 16? I think the inequality you have used is wrong

    Also, Rouche Theorem in my book says
    |f(z) - g(z)| < |f(z)|

    and not
    |f(z) - g(z)| < |f(z)| +|g(z)|
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  4. #4
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    If this is correct and someone can help me to the proof I guess then we would have solved it

    |15z + 1| < |z^5+15z + 1| for all z on circle |z|=1
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  5. #5
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    There are different ways to write Rouche's Theorem. The simplest for me is:

    If f(z) and g(z) are analytic inside and on a simple closed curve C and if |g(z)|<|f(z)| on C, then f and f+g have the same number of zeros in C.

    So let f(z)=15z and g(z)=z^5+1 and C be the unit circle. Then |f(z)|>|g(z)| on C so therefore f(z)=15z and f+g=z^5+15z+1 have the same number of zeros inside C which is one.
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  6. #6
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    Quote Originally Posted by shawsend View Post
    There are different ways to write Rouche's Theorem. The simplest for me is:

    If f(z) and g(z) are analytic inside and on a simple closed curve C and if |g(z)|<|f(z)| on C, then f and f+g have the same number of zeros in C.

    So let f(z)=15z and g(z)=z^5+1 and C be the unit circle. Then |f(z)|>|g(z)| on C so therefore f(z)=z and f+g=z^5+15z+1 have the same number of zeros inside C which is one.
    Thanks. This looks correct shawsend

    However am I correct in saying the following:

    for all on circle

    Because this will imply we have 5 zeros. In this I have used
    f(z)=z^5+15z+1 and g(z) = -15z-1
    (This is to bring the Rouche in-equality in the form you wrote ( |g(z)|<|f(z)|)

    If that is the case (which I think is) then there is some contradiction in what you have proved (that there is just one zero) and what this logic says (that all the 5 zeros are inside the unit circle).

    Where am I messing up?
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  7. #7
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    I don't think your inequality is valid. I used the code below to draw the zeros. It shows one zero in the unit circle (red dot are zeros).

    Code:
    myEqn = z^5 + 15 z + 1;
    roots = N[z /. Solve[myEqn == 0, z]]
    points = Point@ {Re[#], Im[#]} & /@ roots
    Show[Graphics[{{PointSize[0.01], Red, points}, {Blue, 
        Circle[{0, 0}, 1]}}, PlotRange -> {{-2, 2}, {-2, 2}}, 
      Axes -> True]]
    Attached Thumbnails Attached Thumbnails Question on Rouche Theorem -1-arouche-plot.jpg  
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  8. #8
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    Thanks very much shawsend !!!
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