I'll preface this by saying that this just isn't getting through to me. I know the material, but my brain feels like it doesn't, for lack of a better word, "fit."(adsbygoogle = window.adsbygoogle || []).push({});

Looking back at Riemann surfaces in complex analysis after familiarizing myself with some differential geometry really makes them more confusing. Or, at least, my understanding clashes with what the internet seems to think.

One of the uses of Riemann surfaces is to be able to define multivalued functions between them. I'm having a lot of trouble understanding the visual aspect of them. Many of the sources I go to seem to think that a Riemann surface can be visualized as a singular embedded surface in ##\mathbb{R}^3##. Shouldn't it instead be 2 surfaces, or a surface in ##\mathbb{R}^4##?

Also, there is the problem of defining functions between them. This is my primary problem. Let's take, for example, the complex logarithm. Wolfram Alpha gives 2 surfaces, which I imagine correspond with the real and imaginary parts of ##\ln(z)##. If we imagine it as a single valued function using a Riemann surface ##S##, are we defining a map ##\ln_S: \mathbb{C}\to S##, or a map ##\ln_S: S\to\mathbb{C}##? If it's the first one, are the surfaces shown by Wolfram just the real and imaginary parts of the image of ##\mathbb{C}## under ##\ln_S##? If it's the second one...what the heck is ##S##?

I don't understand, and I've managed to give myself something of a headache. Yay, math? :tongue:

Any help is greatly appreciated.

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# Using Riemann Surfaces

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