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algebraic equation; Cauchy transform; quadratic differential
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.
[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 10.1016/j.jmaa.2014.02.040 | MR 3182748 | Zbl 1295.30015
[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 0096806 | Zbl 0083.29606
[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 10.1007/s00365-003-0536-3 | MR 2078083 | Zbl 1069.33008
[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 10.1016/S0377-0427(00)00654-3 | MR 1858305 | Zbl 0990.33009
[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 0507768 | Zbl 0298.30014
[6] Pritsker, I. E.: How to find a measure from its potential. Comput. Methods Funct. Theory 8 597-614 (2008). DOI 10.1007/BF03321707 | MR 2419497 | Zbl 1160.31004
[7] Strebel, K.: Quadratic Differentials. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) Vol. 5 Springer, Berlin (1984). MR 0743423 | Zbl 0547.30038
[8] Vasil'ev, A.: Moduli of Families of Curves for Conformal and Quasiconformal Mappings. Lecture Notes in Mathematics 1788 Springer, Berlin (2002). DOI 10.1007/b83857 | MR 1929066 | Zbl 0999.30001
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