Riemann surfaces from weierstrass's idea (as a complete analytic function)

Question

Currently I am reading some textbooks on Riemann surfaces, mainly Donaldson's book. In Donaldson's book (and some other quite newly wrote book), most Riemann surfaces seem constructed in a "algebraic curve" style or some quotient space style, which seems quite geometric.

But in some older books like Ahlfors's book Complex Analysis or George Springer's introduction to riemann surfaces, they give a constructive view of Riemann surfaces via "germ/sheaf of analytic functions" or the complete analytic. In this way it seems quite easier to handle some "transcendental" function like $w=\sqrt{z} + \log(z)$.

And some related questions I found:

• About the Riemann surface associated to an analytic germ
• Riemann Surface Comprehension

So maybe my question may be sumarized to:

1. Is there any newly textbook view Riemann surfaces comprehensively from this point? (both books of Ahfors and George Spring are quite old)
2. Why most modern textbook on Riemann surfaces does not discuss this view anymore? How do we handle Riemann surface like $w=\sqrt{z} + \log(z)$ in a modern view?
3. Is it because that the complete analytic function is quite hard to do computation? I do understand it's hard to use power series to do analytic continuation. But it seems not that hard to do the "instructive" analytic continuation.

Sorry for the question being kind of soft, thanks!

Answer 1

1. I do not think so.

2 and 3: This is mostly a matter of fashion and taste of people writing modern textbooks. The modern books on Riemann surfaces are mostly written by geometers rather than analysts, which is reflected in the resulting neglect of multivalued functions. As as example of using the "modern" (geometric or topological if you wish) viewpoint on Riemann surfaces for answering questions dealing with multivalued functions, check my answer here on the treatment of multivalued function from the viewpoint of covering spaces.

I remember answering an MSE request for references regarding multivalued functions from Riemann surface viewpoint awhile ago, I just cannot find it. My recollection is that I also settled on Ahlfors and Springer. Maybe also Forster, relating to germs of analytic functions, but his book is also old.