Geodesic
Closed Formula for the Capacity of Several Aligned Segments
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and its applications, Tome 298 (2017), pp. 67-74.

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We present a universal closed formula in terms of theta functions for the Log-capacity of several segments on a line. The formula for two segments was obtained by N. Achieser (1930); three segments were considered by T. Falliero and A. Sebbar (2001).
Keywords: capacity, theta functions, closed formula. 
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A. B. Bogatyrev; O. A. Grigoriev. Closed Formula for the Capacity of Several Aligned Segments. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and its applications, Tome 298 (2017), pp. 67-74. http://geodesic.mathdoc.fr/item/TM_2017_298_a4/

[1] Achieser N., “Sur les polynomes de Tchebyscheff pour deux segments”, C. r. Acad. sci. Paris, 191 (1930), 754–756 | Zbl

[2] Aptekarev A. I., “Asimptoticheskie svoistva mnogochlenov, ortogonalnykh na sisteme konturov, i periodicheskie dvizheniya tsepochek Toda”, Mat. sb., 125:2 (1984), 231–258 | MR

[3] Betsakos D., Samuelsson K., Vuorinen M., “The computation of capacity of planar condensers”, Publ. Inst. math., 75 (2004), 233–252 | DOI | MR | Zbl

[4] Bogatyrev A. B., “Elementarnaya konstruktsiya shtrebelevykh differentsialov”, Mat. zametki, 91:1 (2012), 143–146 | DOI | Zbl

[5] Bogatyrev A. B., “Konformnoe otobrazhenie pryamougolnykh semiugolnikov”, Mat. sb., 203:12 (2012), 35–56 | DOI | Zbl

[6] Bogatyrev A., Extremal polynomials and Riemann surfaces, Springer Monogr. Math., Springer, Berlin, 2012 | DOI | MR | Zbl

[7] Bogatyrev A. B., “Image of Abel–Jacobi map for hyperelliptic genus 3 and 4 curves”, J. Approx. Theory, 191 (2015), 38–45 | DOI | MR | Zbl

[8] Bogatyrev A. B., Grigor'ev O. A., “Conformal mapping of rectangular heptagons. II”, Comput. Methods Funct. Theory (to appear)

[9] Dubinin V. N., Karp D. B., “Generalized condensers and distortion theorems for conformal mappings of planar domains”, The interaction of analysis and geometry, Contemp. Math., 424, Amer. Math. Soc., Providence, RI, 2007, 33–51 | DOI | MR | Zbl

[10] Falliero T., Sebbar A., “Capacité d'une union de trois intervalles et fonctions thêta de genre 2”, J. math. pures appl., 80:4 (2001), 409–443 | DOI | MR | Zbl

[11] Farkas H. M., Kra I., Riemann surfaces, Grad. Texts Math., 71, Springer, New York, 1980 | DOI | MR | Zbl

[12] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[13] Grigorev O. A., “Chislenno-analiticheskii metod konformnogo otobrazheniya mnogougolnikov s shestyu pryamymi uglami”, ZhVM i MF, 53:10 (2013), 1629–1638 | MR | Zbl

[14] Lukashov A. L., Peherstorfer F., “Automorphic orthogonal and extremal polynomials”, Can. J. Math., 55 (2003), 576–608 | DOI | MR | Zbl

[15] Mumford D., Tata lectures on theta, v. II, Birkhaüser, Boston, 1983 ; 1984; 1991 | MR | Zbl

[16] Rauch H. E., Farkas H. M., Theta functions with applications to Riemann surfaces, Williams Wilkins Co., Baltimore, 1974 | MR | Zbl

[17] Rumely R. S., Capacity theory on algebraic curves, Lect. Notes Math., 1378, Springer, Berlin, 1989 | DOI | MR | Zbl

[18] Widom H., “Extremal polynomials associated with a system of curves in the complex plane”, Adv. Math., 3:2 (1969), 127–232 | DOI | MR | Zbl