We consider a functional equation of the form f(qz) = p(z)f(z), z ∈ ℂ\{0}, q ∈ ℂ\{0}, |q| < 1. For some fixed elementary functions p(z), we determine holomorphic solutions of this equation. These solutions are generalizations of loxodromic functions. Some solutions can be represented via the Schottky–Klein prime function.
This is a preview of subscription content, log in via an institution to check access.
References
D. G. Crowdy, “Geometric function theory: a modern view of a classical subject,” Nonlinearity, 21, No. 10, T205–T219 (2008).
J. Marcotte and M. Salomone, “Loxodromic spirals in M. C. Escher’s sphere surface,” J. Humanist. Math., 4, No. 2, 25–46 (2014).
Y. Hellegouarch, Invitation to the Mathematics of Fermat–Wiles, Academic Press, San Diego (2002).
O. Hushchak and A. Kondratyuk, “The Julia exceptionality of loxodromic meromorphic functions,” Visn. Lviv Univ., Ser. Mech., Mat., 78, 35–41 (2013).
V. S. Khoroshchak, A. Ya. Khrystiyanyn, and D. V. Lukivska, “A class of Julia exceptional functions,” Carpath. Math. Publ., 8, No. 1, 172–180 (2016).
V. S. Khoroshchak and A. A. Kondratyuk, “The Riesz measures and a representation of multiplicatively periodic 𝛿-subharmonic functions in a punctured Euclidean space,” Mat. Stud., 43, No. 1, 61–65 (2015).
V. S. Khoroshchak and N. B. Sokulska, “Multiplicatively periodic meromorphic functions in the upper half plane,” Mat. Stud., 42, No. 2, 143–148 (2014).
V. S. Khoroshchak and A. A. Kondratyuk, “Stationary harmonic functions on homogeneous spaces,” Ufimsk. Mat. Zh., 7, No. 4, 155–159 (2015).
A. Ya. Khrystiyanyn and A. A. Kondratyuk, “Meromorphic mappings of torus onto the Riemann sphere,” Carpath. Math. Publ., 4, No. 1, 155–159 (2012).
A. Ya. Khrystiyanyn and A. A. Kondratyuk, “Modulo-loxodromic meromorphic function in C\0,” Ufimsk. Mat. Zh., 8, No. 4, 156–162 (2016).
F. Klein, “Zur Theorie der Abel’schen Functionen,” Math. Ann., 36, 1–83 (1890).
A. A. Kondratyuk and V. S. Zaborovska, “Multiplicatively periodic subharmonic functions in the punctured Euclidean space,” Mat. Stud., 40, No. 2, 159–164 (2013).
A. A. Kondratyuk, “Loxodromic meromorphic and 𝛿-subharmonic functions,” in: Proc. of the Workshop on Complex Analysis and Its Applications to Differential and Functional Equations (Joensuu, Finland, August 20–22, 2014), University of Eastern Finland, 14 (2014), pp. 89–99.
O. Rausenberger, Lehrbuch der Theorie der Periodischen Functionen Einer Variabeln, Teubner, Leipzig (1884).
S. Kos and T. K. Pogány, “On the mathematics of navigational calculations for meridian sailing,” Electron. J. Geography Math. (2012).
F. Schottky, “Über eine specielle Function welche bei einer bestimmten linearen Transformation ihres Arguments unvera¨ndert bleibt,” J. Reine Angew. Math., 101, 227–272 (1887).
G. Valiron, Cours d’Analyse Mathematique, Theorie des Fonctions, Masson, Paris (1947).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 9, pp. 1284–1288, September, 2017.
Rights and permissions
About this article
Cite this article
Lukivs’ka, D.V. Some Holomorphic Generalizations of Loxodromic Functions. Ukr Math J 69, 1490–1495 (2018). https://doi.org/10.1007/s11253-018-1449-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-018-1449-4