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Thread: Solved Complex Analysis

  1. #1
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    Solved Complex Analysis

    $\displaystyle \int\frac{sin(\pi.z)}{z^{2}(z-2)}dz$

    This was the the problem i was give and it asks to integrate around the contours center a=-2, a=0, a=2 and radius 1.

    i) a=-2 gives and answer of 0 because it is holomorphic everywhere

    ii) a=0 gives and answer of $\displaystyle \pi i$

    iii) a=2 gives and answer of 0 as $\displaystyle \int \frac{f(z)}{z-2} dz$, when $\displaystyle f(z)=\frac{sin(\pi.z)}{z^{2}}$, = $\displaystyle 2\pi.i \frac{sin(\pi.2)}{2^{2}}=0$

    could some please check my working for me as am a bit confused about $\displaystyle sin(\pi.z)$ around the contours.

    thanks bobisback
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  2. #2
    MHF Contributor chisigma's Avatar
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    The essential point is that the integral can be written as...

    $\displaystyle \int_{\gamma}\frac{\sin (\pi z)}{z^{2}\cdot (z-2)}\cdot dz= \pi \int_{\gamma} \frac{sinc (z)}{z\cdot (z-2)}\cdot dz$ (1)

    ... where $\displaystyle sinc (z) = \frac{\sin (\pi z)}{\pi z}$ is an entire function [i.e. a fuction which is analytic in the whole complex plane...] , so that the function to be integrate has only single poles in $\displaystyle z=0$ and $\displaystyle z=2$. In this case we can apply the Cauchy's integral formula...

    $\displaystyle \int_{\gamma} f(z)\cdot dz = 2\pi j \sum_{n} r_{n}$ (2)

    ... where the $\displaystyle r_{n}$ are the residues of all single poles inside $\displaystyle \gamma$. Now we have three possibilities...

    a) if $\displaystyle \gamma$ is the unit circle centered in $\displaystyle a=-2$ there are no poles inside $\displaystyle \gamma$ and is $\displaystyle \int_{\gamma} f(z)\cdot dz =0$...

    b) if $\displaystyle \gamma$ is the unit circle centered in $\displaystyle a=0$ there is a pole in $\displaystyle z=0$ inside $\displaystyle \gamma$ and its residue is $\displaystyle \lim_{z \rightarrow 0} \pi \frac{sinc (z)}{z-2} = - \frac{\pi}{2}$ so that is $\displaystyle \int_{\gamma} f(z)\cdot dz = - j \pi^{2}$...

    c) if $\displaystyle \gamma$ is the unit circle centered in $\displaystyle a=2$ there is a pole in $\displaystyle z=2$ inside $\displaystyle \gamma$ and its residue is $\displaystyle \lim_{z \rightarrow 2} \pi \frac{sinc (z)}{z} = 0$ so that is $\displaystyle \int_{\gamma} f(z)\cdot dz = 0$...

    Kind regards

    $\displaystyle \chi$ $\displaystyle \sigma$
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