Timeline for Expansion around a singular point of a multivalued meromorphic function (due to Riemann/Cauchy)
Current License: CC BY-SA 4.0
15 events
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| Jun 24, 2021 at 0:50 | comment | added | user267839 | @DrPotato: The only googleble free available version is the original paper 'Theorie der Abel'schen Functionen'. See there page 17 at the end of first subchapter of 'Erste Abtheilung'. | |
| Jun 22, 2021 at 19:56 | comment | added | Dr Potato | Please include some reference where we can see the context and statement of this theorem. | |
| Mar 26, 2021 at 1:41 | comment | added | user267839 | I found the answer: In previous subpart of his Part I in the quoted paper Riemann shortly remarks that he want to deal with single-valued functions on RS $T$ with isolated singularities (so moromorphics $T \to P^1$) *and their integrals (!) That's why $log \ r$ term in the series may occure which confused me. | |
| Mar 23, 2021 at 5:12 | comment | added | user267839 | @Will Sawin: That's a good suggestion, possibly I somewhere overlooked that Riemann maybe somewhere explicitely assumed that he deals only with multi-valued functions with single-valued derivative. I don't know the historical background but possibly Riemann studied during his lifetime when he 'invented' Riemann surfaces only multi-valued functions with single-valued derivative, I don't known, but of course the expansion above make is always given only with multi-valued functions with single-valued derivative. | |
| Mar 21, 2021 at 15:23 | comment | added | Will Sawin | Perhaps, in context, he has reason to assume the derivative of the function is single-valued (and has no essential singularity)? Then the derivative would have a Laurent series expansion, which, integrating, would give an expansion of this form. | |
| Mar 20, 2021 at 18:20 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor Math Jaxing (${log}$ substituted by $\log$), formatting and typo fixing.
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| Mar 20, 2021 at 17:24 | history | edited | user267839 | CC BY-SA 4.0 |
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| Mar 20, 2021 at 1:58 | history | edited | user267839 | CC BY-SA 4.0 |
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| Mar 20, 2021 at 1:49 | review | Close votes | |||
| Mar 29, 2021 at 3:08 | |||||
| Mar 20, 2021 at 1:49 | comment | added | user267839 | But concerning this question I don't agree with you. I looked in several books on this topic and I nowhere found a treating of the problem I described above. Honestly, I don't think that this treating of the singularity using expansion containing this log term & it's meaning there is standard and something 'that can be easy translated in modern language'. If you think that I wrong then I would appreciate if you could give a hint where I could look up it. | |
| Mar 20, 2021 at 1:48 | comment | added | user267839 | @AlexandreEremenko: maybe I'm running to overestimate my knowledge on this topic, but I think that the most principles which one can look up in a basic modern complex variables book are at least roughly familar to me. So I know at least basic notions. About the advise you gave me in my last question here mathoverflow.net/questions/386700/… you are right, that was almost basic. A local parameter on a RS is exactly what Riemanns calls infinitely small function of first order. | |
| Mar 20, 2021 at 1:31 | comment | added | Alexandre Eremenko | I repeat my advise: learn the basic notions from some modern Complex variables book, and only AFTER that read Riemann. His terminology and the manner of exposition are out of date. Once you have some experience with the subject, it is easy to translate what he says to the modern language. Riemann did not write for the beginners. And his terminology is outdated. It is not an easy reading for a modern student. | |
| Mar 20, 2021 at 1:09 | history | edited | user267839 | CC BY-SA 4.0 |
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| Mar 19, 2021 at 23:39 | history | edited | user267839 | CC BY-SA 4.0 |
edited body
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| Mar 19, 2021 at 23:23 | history | asked | user267839 | CC BY-SA 4.0 |