I have 3 questions which I have some idea how to solve it, but need confirmations on.
1) Sketch the set of continuity of the function given by the conditions:
f(z) = z if |z| < or = 1
|z|^3 if |z|>1
Is this set open? Is it connected?
So, the sketch should have a circle with radius 1 right? And it is NOT connected since f(z) does not equal at z=1 (or as lim --->1)? And it is open?
2) Let f(z) = z/(z^2 + 1) + e^(1/z)
Find and classify all isolated singularities, and compute the residues at each singularity.
Left polynomial part has removable singularities of +,-i and right has removable singularities at 0, and order of pole is all 1.... is that correct? Just wondering what would be the appropriate justification for it... How should I explain why those are the isolated singularities?
3) What is the order of zero of z^3(1-cos^2(z)) / (e^z-1) at 0
I'm actually not so sure how to do this one. I don't think I should start deriving, so do I just go 1/f and find the order of poles? Would the order of pole at 0 be 4? since we have 3 for z^3 and one for (1-cos^2(z))?
Any help on those questions would be very appreciated. ^^