Abstract
We consider the classical Chazy equation, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. A special solution was found in the upper half plane H with the same tessellation of H as that of the modular group. This allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. A global solution to Chazy equation in elliptic and theta functions was found that allows parametrization of an arbitrary solution to Chazy equation. The results have applications to analytic number theory.
Similar content being viewed by others
References
B. C. Berndt, Number Theory in the Spirit of Ramanujan (Am. Math. Soc., Providence, 2006).
J. Chazy, “Sur les équations différentielles dont l’intégrale générale est uniforme et admet des singularitiés essentielles mobiles,” C.R. Acad. Sci. Paris 149, 563–565 (1909).
G. Halphen, “Sur une systéme d'équations différentielles,” C.R. Acad. Sci. Paris 92, 1101–1103 (1881).
M. J. Ablowitz, S. Chakravarty, and H. Halm, “Integrable systems and modular forms of level 2,” J. Phys. A: Math. Gen. 39, 15341–15353 (2006).
P. A. Glarkson and P. J. Olver, “Symmetry and the Chazy equation,” J. Differ. Equations 124, 225–246 (1996).
H. Blasius, “Grenzschichten in Flüssigkeiten mit kleiner Reibung,” Z. Math. Phys. 56, 1–37 (1908).
J. P. Boyd, “The Blasius function in the complex plane,” Experiment. Math. 8, 381–394 (1999).
V. P. Varin, “A solution of the Blasius problem,” Comput. Math. Math. Phys. 54 (6), 1025–1036 (2014).
M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and the Inverse Scattering, Lect. Notes Math., Vol. 149 (Cambridge Univ. Press, Cambridge, 1991).
Z. Nehari, Conformal Mapping (McGraw-Hill, New York, 1952).
Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol.1.
T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory (Springer-Verlag, New York, 1990).
N. Joshi and M. D. Kruskal, “A local asymptotic method of seeing the natural barrier of the solutions of the Chazy equation,” in Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Ed. by P. A. Clarkson, NATO ASI Ser. C: Math. Phys. Sci., Vol. 413 (Kluwer, Dordrecht, 1992).
M. D. Kruskal, N. Joshi, and R. Halburd, “Analytic and asymptotic methods for nonlinear singularity analysis: A review and extensions of tests for the Painlevé property,” in Integrability of Nonlinear Systems, Ed. by Y. Kosmann-Schwarzbach (Springer, Berlin, 2004).
Sloane Online Encyclopedia of Integer Sequences. http://oeis.org/wiki/Sum_of_divisors_function.
C. Carathéodory, Theory of Functions of a Complex Variable (Chelsea, New York, 1954), Vol.2.
S. Chakravarty and M. J. Ablowitz, Parameterizations of the Chazy Equation. http://arxiv.org/abs/0902.3468v1.
D. Zagier, “Elliptic modular forms and their applications”, in The 1-2-3 on Modular Forms, Ed. by J. H. Bruinier et al. (Springer, Berlin, 2008).
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
S. Ramanujan, “On certain arithmetical functions,” Trans. Camb. Philos. Soc. 22, 159–184 (1916); in Collected Papers of Srinivasa Ramanujan, Ed. by G. H. Hardy et al. (Cambridge Univ. Press, Cambridge, 1927).
P. C. Toh, “Differential equations satisfied by Eisenstein series of level 2,” Ramanujan J. 25, 179–194 (2011).
J. G. Huard et al., “Elementary evaluation of certain convolution sums involving divisor functions,” in Number Theory for the Millennium II, Ed. by M. A. Bennett (A. K. Peters, Natick, Mass., 2002), pp. 229–274.
J. C. Lagarias, “An elementary problem equivalent to the Riemann hypothesis,” Math. Mon. 109 (6), 534–543 (2002).
V. P. Varin, “Flat expansions and their applications,” Comput. Math. Math. Phys. 55 (5), 797–810 (2015).
Y. V. Nesterenko, “Algebraic independence for values of Ramanujan functions,” in Introduction to Algebraic Independence Theory, Ed. by Y. V. Nesterenko and P. Philippon (Springer, Berlin, 2001).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic, New York, 2007).
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge Univ. Press, Cambridge, 1927).
C. G. J. Jacobi, “Fundamenta Nova Theoriae Functionum Ellipticarum,” in Jacobi’s Gesammelte Werke (Chelsea, New York, 1969).
G. Robin, “Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann,” J. Math. Pures Appl. Neuv. Ser. 63 (2), 187–213 (1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2017, Vol. 57, No. 2, pp. 210–236.
The article was translated by the author.
Rights and permissions
About this article
Cite this article
Varin, V.P. Special solutions to Chazy equation. Comput. Math. and Math. Phys. 57, 211–235 (2017). https://doi.org/10.1134/S0965542517020154
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542517020154