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One of the way to motivate the definition of a Riemann surface is to consider a natural domain for the so called multivalued complex functions like the logarithm.We can in fact construct Riemann surface $X$ such that $p:X\to \mathbb C^\times$ is a topological covering and branches of the logarithms glued together forming a single valued continuous logarithm on the domain $X$.Now if we consider $\log:X\to \mathbb C$ would that be a homeomorphism onto $\mathbb C$ or $\mathbb C\setminus \{0\}$.Can we also say $X$ is analytically isomorphic to $\mathbb C$?

Can someone please help me to sort this thing out.I am really confused with this.I am a beginner and I might get stuck with petty things.

Ajin Shaji Jose
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Kishalay Sarkar
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    As Andrew Huang said, your patched-together log is a homeomorphism of $X$ onto $\mathbb C$. Note that this is very different from $p$, which is not one-to-one and which maps onto $\mathbb C-{0}$. Also, $X$ has a structure of analytic manifold, which makes log an analytic isomorphism and also makes $p$ analytic (and locally an isomorphism). People usually define this analytic structure on $X$ as the unique one that makes $p$ a local analytic isomorphism; it is then a theorem that the same analytic structure makes log an analytic isomorphism. – Andreas Blass Aug 06 '24 at 17:46

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tl; dr: The Riemann surface of $\log$ is the complex line $\mathbf{C}$.

In the complex Cartesian plane with coordinates $(z, w)$, the locus $w = \exp z$ "is" the exponential function viewed as a function of $z$, and "is" the total logarithm viewed as a "function of $w$." The mapping $z \mapsto(z, \exp z)$, defined on $\mathbf{C}$, is a biholomorphism onto its image. (The image here may be helpful.)

Andrew D. Hwang
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