Geodesic
Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 351-363
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-016-0260-3
Classification : 28A99, 30L05
Keywords: algebraic equation; Cauchy transform; quadratic differential 
@article{10_1007_s10587_016_0260_3,
     author = {Atia, Mohamed Jalel and Thabet, Faouzi},
     title = {Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic {Cauchy} transform},
     journal = {Czechoslovak Mathematical Journal},
     pages = {351--363},
     year = {2016},
     volume = {66},
     number = {2},
     doi = {10.1007/s10587-016-0260-3},
     mrnumber = {3519606},
     zbl = {06604471},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0260-3/}
}
TY  - JOUR
AU  - Atia, Mohamed Jalel
AU  - Thabet, Faouzi
TI  - Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform
JO  - Czechoslovak Mathematical Journal
PY  - 2016
SP  - 351
EP  - 363
VL  - 66
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0260-3/
DO  - 10.1007/s10587-016-0260-3
LA  - en
ID  - 10_1007_s10587_016_0260_3
ER  - 
%0 Journal Article
%A Atia, Mohamed Jalel
%A Thabet, Faouzi
%T Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform
%J Czechoslovak Mathematical Journal
%D 2016
%P 351-363
%V 66
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-016-0260-3/
%R 10.1007/s10587-016-0260-3
%G en
%F 10_1007_s10587_016_0260_3
Atia, Mohamed Jalel; Thabet, Faouzi. Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 351-363. doi: 10.1007/s10587-016-0260-3

[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. | DOI | MR | Zbl

[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). | MR | Zbl

[3] Kuijlaars, A. B. J., McLaughlin, K. T.-R.: Asymptotic zero behavior of Laguerre polynomials with negative parameter. Constructive Approximation 20 (2004), 497-523. | DOI | MR | Zbl

[4] Martínez-Finkelshtein, A., Martínez-González, P., Orive, R.: On asymptotic zero distribution of Laguerre and generalized Bessel polynomials with varying parameters. J. Comput. Appl. Math. 133 (2001), 477-487 Conf. Proc. (Patras, 1999), Elsevier (North-Holland), Amsterdam. | DOI | MR | Zbl

[5] Pommerenke, C.: Univalent Functions. With a Chapter on Quadratic Differentials by Gerd Jensen. Studia Mathematica/Mathematische Lehrbücher. Band 25. Vandenhoeck & Ruprecht, Göttingen (1975). | MR | Zbl

[6] Pritsker, I. E.: How to find a measure from its potential. Comput. Methods Funct. Theory 8 597-614 (2008). | DOI | MR | Zbl

[7] Strebel, K.: Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Vol. 5 Springer, Berlin (1984). | MR | Zbl

[8] Vasil'ev, A.: Moduli of Families of Curves for Conformal and Quasiconformal Mappings. Lecture Notes in Mathematics 1788 Springer, Berlin (2002). | DOI | MR | Zbl

Cité par Sources :