Timeline for Is this a rational function?
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23 events
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| Jun 27, 2017 at 18:18 | comment | added | R.P. | @LasseRempe-Gillen: Oh yes, I see now that you wrote explicitly 'as a function on the Riemann sphere'. My bad, sorry. | |
| Jun 27, 2017 at 17:41 | comment | added | Lasse Rempe | @René A pole is precisely a removable singularity, when the function is considered as a function taking values in the Riemann sphere. | |
| Jun 25, 2017 at 17:33 | comment | added | R.P. | @DuchampGérardH.E.: a reference for the above statement is O. Forster, Lectures on Riemann surfaces, Corollary 2.9. | |
| Jun 25, 2017 at 17:12 | comment | added | R.P. | @LasseRempe-Gillen: I think you meant to write 'pole' instead of 'removable singularity'. The key point is that a meromorphic function which has at worst a pole at $\infty$ in fact defines a meromorphic function on the Riemann sphere, and such a function is known to be rational. | |
| Jun 2, 2015 at 21:52 | comment | added | Lasse Rempe | Any globally defined meromorphic function that has a removable singularity (as a function on the Riemann sphere) at $\infty$ is clearly rational. Hence any non-rational meromorphic function in the plane has an essential singularity at infinity, and is transcendental. | |
| Jun 1, 2015 at 18:54 | comment | added | Duchamp Gérard H. E. | @LasseRempe-Gillen : Could you give a reference for this statement "any globally defined meromorphic function is either rational or transcendental". Thanks in advance. | |
| Jun 1, 2015 at 16:52 | vote | accept | Pablo | ||
| Jun 1, 2015 at 11:41 | comment | added | GH from MO | @LasseRempe-Gillen: I had a similar feeling, but I did not know it from the top of my head, nor did I have the energy to verify it or find the appropriate reference. Thank you! | |
| Jun 1, 2015 at 11:36 | comment | added | Lasse Rempe | @GeraldEdgar - surely this follows also from the first argument given in this answer? Indeed, any globally defined meromorphic function is either rational or transcendental. | |
| Jun 1, 2015 at 5:49 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 31, 2015 at 23:55 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 31, 2015 at 20:50 | comment | added | Gerald Edgar | So remark 2 is even stronger than the question: this function is not algebraic | |
| May 31, 2015 at 20:48 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 31, 2015 at 19:14 | history | edited | GH from MO | CC BY-SA 3.0 |
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| May 31, 2015 at 17:59 | history | answered | GH from MO | CC BY-SA 3.0 |