f(z)=z(ez−1)1
The singularities are z=0 and z=2nπi,n=±1,±2,....
The singularity at z=0 is a pole of order 2 since z=0 is a zero of order 2 of z(ez−1).
z(ez−1)1=z(z+z2/2!+z3/3!+...)1
=z2(1+z/2!+z2/3!+...)1=z21g(z),g(0)=0
The singularities z=2nπi,n=±1,±2,.... are simple poles since they are simple zeros of z(ez−1)1.
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