It is my understanding that Riemann surfaces are introduced in complex analysis in order to deal with "difficult" functions such as $\log$, $(\cdot)^{p/q}$ and the like by changing their domain from $\mathbb C$ to a suitable Riemann surface, which would technically be a 1-dimensional complex manifold with a holomorphic atlas. In this sense, then, Riemann surfaces are to complex numbers what smooth curves are to real numbers.

So why do we not feel the need to introduce smooth curves as "real Riemann surfaces" in the standard single-variable real analysis courses? What advantages would it bring us to define $(\cdot)^{1/2n}$ over $2$ copies of $\mathbb R^+$ stitched together at the origin instead of choosing the "positive branch" as is usually done?

(I haven't thought long enough about this, so please feel free to point out the obvious if needed.)

  • $\begingroup$ $x^{2n}=a$ only has $2$ real solutions, not $2n$. So, the positive and negatives rays attached at the origin (the real line) is enough. That function doesn't give enough motivation. But smooth curves are studied indeed. $\endgroup$ – saltandpepper Mar 30 '18 at 16:57
  • $\begingroup$ @saltandpepper Sorry, typo. Corrected. $\endgroup$ – giobrach Mar 30 '18 at 16:59
  • $\begingroup$ I wouldn't say that smooth curves are "studied", because as abstract manifold they all are isomorphic to disjoint union of circles and lines. $\endgroup$ – xsnl Mar 30 '18 at 18:26

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