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Math Help - Singularities and integrating

  1. #1
    Junior Member
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    Singularities and integrating

    Hi, I have some past papers with no answers and not much idea on how to find them, any help appreciated!!
    For part a) is it something about  z^4 lies on the circle with radius R^4 and so its minimal distance to -sqrt(2)a i s R^2-4a^4?
    part b) use the M-L lemma to show int(f) < \Pi R z e^iz/  R^4-4a^4 which tends to zero as R tends to infinity?
    c) singularities would be z^4=-4a^4 though not sure what these are . Think I may have to brake z^4+4a^4down into composite parts e.g.  (z^2-2ia^2)(z^2+2ia^2)
    part d) I have no idea!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Hint For R sufficiently large we have

    \left |{\dfrac{1}{z^4+4a^4}}\right |\leq \dfrac{1}{R^4-4a^4}

    By a well known property

    \left |{\displaystyle\int_{\Gamma_R}f(z)dz}\right |\leq \dfrac{\pi R}{R^4-4a^4}

    Now, take limits in both sides as R\to +\infty .
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