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Thread: Singularities and integrating

  1. #1
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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$\displaystyle z^4$ lies on the circle with radius $\displaystyle R^4 $and so its minimal distance to -sqrt(2)a i$\displaystyle s R^2-4a^4$?
    part b) use the M-L lemma to show int(f)$\displaystyle <$$\displaystyle \Pi R z e^iz$/$\displaystyle R^4-4a^4$ which tends to zero as $\displaystyle R$ tends to infinity?
    c) singularities would be $\displaystyle z^4=-4a^4 $though not sure what these are . Think I may have to brake $\displaystyle z^4+4a^4$down into composite parts e.g.$\displaystyle (z^2-2ia^2)(z^2+2ia^2)$
    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

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

    By a well known property

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

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