Math Help Forum: Singularities and residues

Find the isolated singularities (if any) of the following function. For each isolated singularity, describe its nature; that is is it removeable or a pole (and of what order) or essential? In each case, calculate the residue of the function at the singularity.

a)
I got a removable singularity at z=0.
but how do I exactly work out the residue for this?

b)
I got none. because you cannot have a complex number which is multiplied to a power of a half. I'm not quite sure about this one...

c)
I got a removable singularity at z=0. Not sure about this one either.... These are some hard functions that I got...

Please help me out!!

for a), should I expand it to Lauren series... and then see what coefficients I get?

Originally Posted by lpd

Find the isolated singularities (if any) of the following function. For each isolated singularity, describe its nature; that is is it removeable or a pole (and of what order) or essential? In each case, calculate the residue of the function at the singularity.

a)
I got a removable singularity at z=0.
but how do I exactly work out the residue for this?

b)
I got none. because you cannot have a complex number which is multiplied to a power of a half. I'm not quite sure about this one...

c)
I got a removable singularity at z=0. Not sure about this one either.... These are some hard functions that I got...

Please help me out!!

a) The residue at a removable singularity is zero. Your job is to think about why ....

b) The function is ....

c) I suggest you try getting the first few terms of the series around z = 0. Is there a 1/z term ....?

Originally Posted by mr fantastic

c) I suggest you try getting the first few terms of the series around z = 0. Is there a 1/z term ....?

theres no 1/z term when i computed the lauren series. does that imply that ?

Originally Posted by lpd

theres no 1/z term when i computed the lauren series. does that imply that ?

Yes.