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Math Help - Complex Analysis: Laurent Series

  1. #1
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    Complex Analysis: Laurent Series

    I'm having some trouble with these two Laurent Series questions:

    1) Consider the Laurent Series expansion of (pi^2-4z^2)/cos(z) which converges on the circle |z|=5. What is the principal part of this expression. What is the largest open set on which the series converges.

    So I have no idea how to begin, especially with the series having to converge on the circle |z|=5. I know we can rewrite the expression as
    (pi-2z)(pi+2z)/cosz. so we have zeros at +/- pi/2. Past that I am lost.

    The second question is similar:
    2) Find the Laurent Series expansion of the function
    (z)/((z^2+1)(9-z^2)
    centered at 0 and convergent at z=2i. What is the largest open set for which this series converges.

    For this one I am confused because I thought we used the Laurent series expansion to fix problems on an annulus.

    Any help?
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  2. #2
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    You do not want to find the zeroes instead you want to find the functions singularities.

    This is when cosz=0.
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  3. #3
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    Quote Originally Posted by whipflip15 View Post
    You do not want to find the zeroes instead you want to find the functions singularities.

    This is when cosz=0.
    Yes this is what I meant to write. At npi/2 there are singularities. Are they removable because of the numerator? How do I set up a series for that.
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  4. #4
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    So I spent some time today on these and wasn't getting anywhere. What really throws me off is the notion that these series converge on a circle instead of at a point or in some domain.
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  5. #5
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    It doesn't mean it must converge 'only' on the circle it is just that the domain includes the circle.

    The second one is easier. It has singularities at i, -i, 3 and -3. It asks for convergence at 2i but the series we want will converge at 1<|z|<3 which include 2i.
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  6. #6
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    Quote Originally Posted by whipflip15 View Post
    It doesn't mean it must converge 'only' on the circle it is just that the domain includes the circle.

    The second one is easier. It has singularities at i, -i, 3 and -3. It asks for convergence at 2i but the series we want will converge at 1<|z|<3 which include 2i.

    So will I come up with different series that converge for |z|<1, 1<|z|<3, and |z|>3? For each of these domains, do I just exclude the factor of the denominator which would give me trouble?
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  7. #7
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    Yes a different solution for each region. To do the second one spit the function into partial fractions then expand each as a geometric sequence (being careful of the radii of convergence.)
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  8. #8
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    Quote Originally Posted by whipflip15 View Post
    Yes a different solution for each region. To do the second one spit the function into partial fractions then expand each as a geometric sequence (being careful of the radii of convergence.)
    So I got this expression writing the things out in partial fractions

    z/20 * ((-1/(1-iz) + (1+(2-iz)+(2-iz)^2+...) + 1/(1-(4+3z)) - (1/9)*(1+(z/3)+(z/3)^2+...))

    Where do I go from here? The two terms which I wrote out in geometric series form don't converge for 1<|z|<3
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