a) Suppose f(z) is analytic and nonzero at a ∈ C, and that g(z) has a non-removable isolated singularity of a given type at a. Show that a is an isolated singular point of fg and find its type.
b) Same question when a is a pole of order N of f.
c) Can anything be asserted about the type of a for fg if f and g have essential singularity at a?
for part a, I do not know how to show that it is still an isolated singular point when you take fg. To find its type, for an essential singularity I am thinking it remains an essential singularity, but I am not sure how to show that.
how is part b not the same thing asked in a? Can't a nonremovable singularity be a pole?
For c, I think the answer is no. Would 3 examples where f*g gives each different type suffice to answer this?