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

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###### Raman Tripathi

User

( 6 months 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 $${z∈C:−π. So then the required holomorphic bijection is just $$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.

Ra
04-Sep-2019

2 Replies

Ra
05-Sep-2019

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11-Sep-2019

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