Holomorphic bijection between C−(−∞,0] and {z∈C:Re(z)>0}

manpreet Best Answer 3 years ago

I'm looking for a holomorphic bijection between C−(−∞,0]C−(−∞,0] and {z∈C:Re(z)>0}{z∈C:Re(z)>0}. I know that LogLog (so the principal value) is a holomorphic bijection between C−(−∞,0]C−(−∞,0] and {z∈C:−π<Im(z)<π}{z∈C:−π. So then the required holomorphic bijection is just exp(12Log(z))=z√exp⁡(12Log(z))=z, 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.