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The only value of this function we can deduce is q_n(0)=0. Moreover any branch cut we take in our complex plane will touch zero. These two facts would make me a bit suspicious that z=0 is the point at which the function can't be continuous except for the fact there is a previous problem which asks the reader to prove a special case of this problem for n=2 in the punctured plane, so at least in that case there must be some other point.

Nevertheless I still have a feeling that the inability for q_n to be continuous will have something to do with having to take some branch cut. Unfortunately despite this being graduate complex analysis the explanation by my professors of the branch cut and the concept of multi-valued functions has always been a bit hand-wavy and thus I don't fully understand how these things are rigorously constructed nor do I fully understand their implications.

Some direction on this problem would be much appreciated, and maybe someone could point me to some online resources that would help me understand the concepts of branch cut and multi-valued function ( wikipedia isn't that helpful ), thanks.