| Title:
|
Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform (English) |
| Author:
|
Atia, Mohamed Jalel |
| Author:
|
Thabet, Faouzi |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
66 |
| Issue:
|
2 |
| Year:
|
2016 |
| Pages:
|
351-363 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We discuss the representability almost everywhere (a.e.) in $\mathbb {C}$ of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials $A(z-a)(z-b)\*(z-c)^{-2} {\rm d} z^{2}$. More precisely, we give a necessary and sufficient condition on the complex numbers $a$ and $b$ for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs. (English) |
| Keyword:
|
algebraic equation |
| Keyword:
|
Cauchy transform |
| Keyword:
|
quadratic differential |
| MSC:
|
28A99 |
| MSC:
|
30L05 |
| idZBL:
|
Zbl 06604471 |
| idMR:
|
MR3519606 |
| DOI:
|
10.1007/s10587-016-0260-3 |
| . |
| Date available:
|
2016-06-16T12:43:02Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145728 |
| . |
| Reference:
|
[1] Atia, M. J., Martínez-Finkelshtein, A., Martínez-González, P., Thabet, F.: Quadratic differentials and asymptotics of Laguerre polynomials with varying complex parameters.J. Math. Anal. Appl. 416 (2014), 52-80. Zbl 1295.30015, MR 3182748, 10.1016/j.jmaa.2014.02.040 |
| Reference:
|
[2] Jenkins, J. A.: Univalent Functions and Conformal Mapping.Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge, Heft 18. Reihe: Moderne Funktionentheorie Springer, Berlin (1958). Zbl 0083.29606, MR 0096806 |
| Reference:
|
[3] Kuijlaars, A. B. J., McLaughlin, K. T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter.Constructive Approximation 20 (2004), 497-523. Zbl 1069.33008, MR 2078083, 10.1007/s00365-003-0536-3 |
| Reference:
|
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| Reference:
|
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| Reference:
|
[6] Pritsker, I. E.: How to find a measure from its potential.Comput. Methods Funct. Theory 8 597-614 (2008). Zbl 1160.31004, MR 2419497, 10.1007/BF03321707 |
| Reference:
|
[7] Strebel, K.: Quadratic Differentials.Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Vol. 5 Springer, Berlin (1984). Zbl 0547.30038, MR 0743423 |
| Reference:
|
[8] Vasil'ev, A.: Moduli of Families of Curves for Conformal and Quasiconformal Mappings.Lecture Notes in Mathematics 1788 Springer, Berlin (2002). Zbl 0999.30001, MR 1929066, 10.1007/b83857 |
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