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Find the poles and/or zeros of the following complex function watch

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    Find the poles and/or zeros of the following complex function:

    f(z)=sinh(z/(z^-1))

    Firstly f(z) can be written as f(z)=sinh(z/((z+1)(z-1)). This function clearly has a zero at z=0 and poles at z=1 and z=-1. However, we would like to know the order of these. The Taylor expansion of f(z):

    f(z)=z/((z+1)(z-1))+z^3/3!((z-1)(z+1))^3+......

    Does this mean the function has an essential singularity at z=1 and z=-1? Also does it mean that there is some kind of infinite order zero? I'm not too sure about the latter, looking at the function I would think its a simple zero since its raised to the power 1 and once differentiated the zero at z=0 no longer exists.

    Thanks.
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    What you've posted as a Taylor expansion isn't actually a Taylor expansion.

    I think it still has essential singularities at +/-1.

    As for what happens at 0, as I understand your argument, it would equally imply that (e^z - 1) has an infinite zero at z=0, which is of course not true.
 
 
 
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