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# some laurent series stuff watch

1. 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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Updated: January 22, 2017
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