A generalization of the complex plane that was originally conceived to make sense of mathematical expressions such as √z or log z. These expressions cannot be made single-valued and analytic in the punctured plane C(0) (that is, the complex plane with the point 0 removed). The difficulty is that for some closed paths the value of the expression when reaching the end of the path is not the same as it is at the beginning. For example, the closed path can be chosen to be the unit circle centered at z = 0 and followed counterclockwise from z = 1. If √z is assigned the value +1 at z = 1, its value at the end of the circuit is −1. Similarly, if log z is assigned the value 0 at z = 1, at the end of the circuit, allowing the values to change continuously, the value is 2πi.