I'm looking for a holomorphic bijection between and . I know that (so the principal value) is a holomorphic bijection between and . So then the required holomorphic bijection is just , where this last expression is the principal value of the complex square root? Does this indeed suffice? Is the principal value of a complex square root always positive? The context: this question appears in my complex analysis syllabus but it seems odd to me that the solution is really this simple. Thanks in advance.