Branch cuts

I understand how to determin eif there is or is not a branch cut needed when there is a value where the p and q are but not in the general case. Please can you explain how this is done please and solve the following question? (See attached file for full problem description)

Complex solutions

I have no understanding how to go about this problem. I am struggling mightily with the course. The problem I am stuck on goes like this: Let A be a complex number and B a real number. Show that the equation |z|^2+Re(Az)+B=0 has a solution if and only if |A|^2 >= 4B. If this is so, show that the solutions set is a circle or a single point. I have no clue about how to begin, or even what |A|^2 has to do with 4B because the equation only calls for the real value of A. Please explain how this works...thank you very much

Couple complex variable questions

(See attached file for full problem description)

Complex Variables

(See attached file for full problem description) I am having trouble with these two problems. Please provide a detailed explanation to both problems. The definition provided is used to help solve both problems. Thank you.

Complex Variables

Please help me to solve parts (a), (b), and (c) of this problem. To solve it, example 2 of section 18 is needed as a guide to show the correct method. I have included the question, and the two pages on which example 2 is show. Page 56 continues the explanation of example 2. Please include an explanation along with the solutions. Thank you for your help.