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Complex analysis: Counting zeros using the argument principle

  1. Apr 13, 2012 #1
    1. The problem statement, all variables and given/known data
    Gamelin VIII.1.6 (8.1.6)
    "For a fixed number a, find the number of solutions of
    [tex]z^5+2z^3-z^2+z=a[/tex] satisfying Re z > 0"

    2. Relevant equations
    The argument principle relating the change in the argument to the number of zeros and poles of the function on the domain.

    3. The attempt at a solution
    This is obviously equivalent to finding the number of zeros of [tex]f(z)= z^5+2z^3-z^2+z-a[/tex]
    So we consider the (open) half circle with radius R in the right half-plane. We break it up in two paths, 1) counterclockwise along the boundary of the half circle and 2) the one along the imaginary axis from R to -R. Along 1) we parametrize by:
    [tex]z=Re^{it}[/tex] where t ranges from -pi/2 to pi/2. Now the z^5 dominates the polynomial, so the increase in argument along this path is approx. equal to the increase in argument along z^5. The change in argument is therefore 5pi along this path.
    Along 2) we parametrize by z=it and get that:
    [tex]f(z) = t^2 - a + it(t^2-1)^2[/tex]. Now obviously we have to consider several cases of the fixed value of a, as the roots of the real and imaginary parts are dependent of a. For a<=0 the change in argument is -pi, so that the total is 4pi => 2 zeros in the domain. for a>0, but different from 1, we get that the change of argument is pi, so that the total is 6pi => 3 zeros. For a=1 I am having problems. As t is positive and large, the values of f(z) is in the first quadrant. As the value of t is negative and large the values are in the fourth quadrant, and since t=-1 is origo we get another -pi/2. Now for t=1 the function maps to origo and so the change is -pi/2. For t=0 we get the point (-1,0), and here I am a bit uncertain of the change in argument.
    1. Is it zero, since both the starting point and ending point are at the real axis?
    2. Is it pi, since it is a counterclockwise rotation along a half circle-ish shape?

    However, both of these reasonings are wrong, for the answer to be correct, I need the contribution of this circle-ish curve to be -2pi. I can't get it to work.
    Also, second question: If one were to consider a horizontal strip-domain. How would one choose to parametrize it?
     
  2. jcsd
  3. May 10, 2012 #2
    Bump/revive from the dead.
     
  4. May 11, 2012 #3
    Why not use Rouche's Theorem?
     
  5. May 11, 2012 #4
    One could of course use Rouche's theorem, but this exercise is given in the section before Rouche's theorem. So it is possible to solve it without. Also, I've dived so deep into this exercise that I really want to know what I've done wrong.
    Thanks for your answer, though!
     
  6. May 12, 2012 #5


    Ok I got an option for you that you may not like: get the answer first and then fit the analysis to it. That means fire up Mathematica and analyze how the argument changes as you tract the contour for varous values of a. But keep in mind the ordinary Arg function in Mathematica only gives the principal value of the argument so that you have to manually code the analytically-continuous change in argument along the path. So do this for a while, get a feel for how it's changing, then try to justify that result through analytic means.
     
  7. May 15, 2012 #6
    I actually solved this "theoretically". What I was doing wrong was using the argument principle where it cannot be used. The problem is that the function has a zero on the path of integration, so to say. If one translates the area by an epsilon value everything works out. :)
     
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