Originally Posted by

**hohoho00** question:

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locate each of the isolated singularities of the given function and tell whether it is removable singularity, a pole, or an essential singularity. If the singularity is removable, give the value of the function at the point; if the singularity is a pole, give the order of the pole.

(e^z-1)/(e^2z-1)

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What I first tried to do was factoring the bottom.

(e^z-1)/((e^z+1)(e^z-1))

but realized

e^z-1 =0 for z = i(2(pi)n) n = 0, 1 , 2 ...

e^z+1=0 for z= i((pi)+2(pi)n) n = 0, 1, 2...

so basically, denominator = 0 whenever z = i(pi)n , n = 0, 1, 2....

I tried to make this into An(z-z0)^n , but was unsuccessful.

But can't really tell if this really is a removable singularity at z = i(pi)n, n= 0, 1, 2.... (I think |f(z)| remains bounded as z --> any of those points, so it is removable?)

I thought about essential singularities, but I don't think that's the case...

If anyone can solve this problem/ explain, that would be of great help. Thanks.