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Math Help - Complex power series and integral question

  1. #1
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    Complex power series and integral question

    Hey,

    I just wanted to see if i was approaching these tute problems correctly, The final two are what i'm struggling with at the moment and I have a little confusion with if cauchys theorem works backwards.


    a) |2+3i|=|2-3i| so false

    b) True since when the coefficients are real the roots come in complex conjugate pairs

    c) Using triangle inequality, 1/|z^2+1|=> 1/|z^2| +2 = 1/ (x^2+y^2+2)
    1/ (x^2+y^2+2)<=1/x^2+2<=1/2

    d)False, cannot be every f since f must be analytic within the domain and curve region

    e) False, i think, by Cauchys integral formula C must be a simple closed curve enclosing Zo, so as C in this question is just a line, False

    f) I think this is true, but i'm not sure if you can use cauchys theorem backwards. So true as int f(z)dz=0 if f is analytic in C. So since int f(z)dz=0, f must be analytic inside and on C

    h) I'm not sure how to approach this, but I was thinking you could just let an be somthing like 1/sqrt(3)^n for the h), then i think it would work, so h) true

    g) i cant see a way to do somthing similar, so i think the answer is false

    Does my logic seem correct?

    Thanks in advance
    Last edited by Daniiel; October 12th 2011 at 04:24 AM.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Complex power series and integral question

    Quote Originally Posted by Daniiel View Post
    Hey,

    I just wanted to see if i was approaching these correctly, The final two are what i'm struggling with at the moment and I have a little confusion with if cauchys theorem works backwards.



    a) |2+3i|=|2-3i| so false

    b) True since when the coefficients are real the roots come in complex conjugate pairs

    c) Using triangle inequality, 1/|z^2+1|=> 1/|z^2| +2 = 1/ (x^2+y^2+2)
    1/ (x^2+y^2+2)<=1/x^2+2<=1/2

    d)False, cannot be every f since f must be analytic within the domain and curve region

    e) False, i think, by Cauchys integral formula C must be a simple closed curve enclosing Zo, so as C in this question is just a line, False

    f) I think this is true, but i'm not sure if you can use cauchys theorem backwards. So true as int f(z)dz=0 if f is analytic in C. So since int f(z)dz=0, f must be analytic inside and on C

    g) and h) I'm not sure how to approach this, but I was thinking you could just let an be somthing like 1/sqrt(3)^n for the h), then i think it would work, so h) true
    but for g) i cant see a way to do somthing similar, so i think the answer is false

    Does my working seem correct?

    Thanks in advance
    Hey, this isn't for a graded assignment, is it?
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  3. #3
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    Re: Complex power series and integral question

    Hey,
    this is for a tutorial class on Thursday,
    They aren't graded but they are 'enforced',
    like if you don't attend a certain amount without reason you get booted
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  4. #4
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    Re: Complex power series and integral question

    Solved
    Last edited by Daniiel; October 12th 2011 at 04:24 AM.
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