Proving the simple poles

How to prove that this two functions have simple poles:

at

at where k is an integer

Regards.

Well, what is the definition of "simple pole"?

you have a simple pole when the maximum power of 1/z you have is 1 in the laurent series of the function

I'm not sure I understood your definition of a 'simple pole'. Is not a pole commonly defined as the values of such that the denominator is zero? And perhaps by simple, you mean that the pole has multiplicity one?

For example, the first function would have a pole when , or equivalently . Then certainly is a pole, but since is periodic, there most certainly are other poles also.