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some laurent series stuff Watch

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    been long time since this was done so just want to check my answers with anyone kind enough to have a look:

    locate and classify the singularities of

    f(z)=1/(z(1-cosh z))


    so ive said have singularities at z=o and z=2kpi(I)

    at z I have got pole of order 3

    since f(z)=1/{z[z^2/2+z^4/4!+..}

    =g(z)/z^3 where g(z) not 0 and z=0

    similarly I got poles of order 2 for the other zeroes of f

    2)

    let f(z)=z sinh(1/(z+1))

    I) find Laurent series about -1

    ii) write down a punctured open disc containing the circle C={z;|Z+1|+1} on whichf is represented by this series

    ii state nature of the singularity of f at -1

    iv)
    integrate f(z) around the circle C

    so I got a Laurent series with even terms

    a_{2n}=-1/[(2n-1)!(z+1)^(n-1)]

    and odd terms

    1/[((2n+1)!(z+1)^(n-1)]

    for ii do they just want 0<|Z+1|<1??

    for iii) looks like essential singularity

    and the integral is just 2(pi)I(-1)
 
 
 
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