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Math Help - please help: Find all the poles and multiplicities of..

  1. #1
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    please help: Find all the poles and multiplicities of..

    Find all the poles and multiplicities of

    a) \left(\frac{4z+6}{z^2+6z+9}\right)^3

    b) \frac{6z}{z^2+5z+6}+\frac{1}{z+2}


    Attempt:

    a) ok so \left(\frac{4z+6}{z^2+6z+9}\right)^3

    = \left(\frac{2z+3}{(z+3)^2}\right)^3

    Would that mean pole at z= -3 of order 2 or order 6?


    b) \frac{6z}{z^2+5z+6}+\frac{1}{z+2}

    = \frac{6z}{(z+2)(z+3)}+ \frac{1}{z+2}

    So there is a pole at z=-3 of order 1 but is the pole at z=-2 of order 1 or order 2?
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  2. #2
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    Re: please help: Find all the poles and multiplicities of..

    For part a) this should help

    \frac{1}{(z-1)^4}

    has pole of order 4. What's the order of the pole of this

    \left(\frac{1}{(z-1)^2}\right)^2?

    For part b) you are not done. What's the definition of pole? Does your function look like the definition in its current form? I mean, what you have is right but it's more technically correct to write it all out as a fraction of polynomials.
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  3. #3
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    Re: please help: Find all the poles and multiplicities of..

    thats what i was asking :P
    We have covered the first example you showed, but not where the whole fraction is raised to a power. My feeling is that it would also be 4 because they are equivalent.

    oh so for part b, 7z+3/ (z+2)(z+3) and then pole at z= -2,-3 both of order 1?

    Thanks for the speedy reply
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  4. #4
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    Re: please help: Find all the poles and multiplicities of..

    Yes, for part b it's like that.

    Your feeling is right. The pole multiplicities are the same. Do you know the limit definition of a pole of order n?
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