Abstract
We find that theta functions are solutions of a fractional partial differential equation that generalizes the diffusion equation. This equation is the limit of a sequence of differential equations for the partial sums of theta functions where the fractional derivatives are given as differentiation matrices for trigonometric polynomials in their Fourier representation, i.e., given as similarities of diagonal matrices under the ordinary discrete Fourier transform. This fact enables the fast numerical computation of fractional partial derivatives of theta functions and elliptic integrals.
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References
Campos, R.G., Meneses, C.: Differentiation matrices for meromorphic functions. Bol. Soc. Mat. Mexicana 12, 121–132 (2006)
Campos, R.G., Pimentel, L.O.: A finite-dimensional representation of the quantum angular momentum operator. Il Nuovo Cimento 116B, 31–45 (2001)
Campos, R.G., Ruiz, R.G.: Fast integration of one-dimensional boundary value problems. Int. J. Mod. Phys. C 24 (2013). https://doi.org/10.1142/S0129183113500824
Campos, R.G., López-López, J.L., Vera, R.: Lattice calculations on the spectrum of Dirac and Dirac-Kähler operators. Int. J. Mod. Phys. 23, 1029–1038 (2008)
Chapront, J., Simon, J.L.: Planetary theories with the aid of the expansions of elliptic functions. Celest. Mech. Dyn. Astron. 63, 171–188 (1996)
Dubrovin, B.A.: Theta functions and non-linear equations. Russ. Math. Surv. 36, 11–92 (1981)
Erdélyi, A. (ed.): Higher Transcendental Functions, vol. II. McGraw Hill, New York (1953)
Golub, G.H., Van Loan, C.F.: Matrix Computations. The Johns Hopkins University Press, Baltimore (1996)
Hankin, R.K.S.: Introducing elliptic, an R package for elliptic and modular functions. J. Stat. Softw. 15, 1–22 (2006)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lawden, D.F.: Elliptic Functions and Applications. Springer Science+Business Media, New York (1989)
Markushevich, A.I.: Theory of Functions of a Complex Variable. AMS Chelsea Publishing, Rhode Island (2011)
McKean, H., Moll, V.: Elliptic Curves. Cambridge University Press, Cambridge (1999)
Meyer, K.R.: Jacobi elliptic functions from a dynamical systems point of view. Am. Math. Mon. 108, 729–737 (2001)
Mumford, D.: Tata Lectures on Theta II. Jacobian Theta Functions and Differential Equations. Birkäuser, Boston (1984)
Mumford, D.: Tata Lectures on Theta I. Birkäuser, Boston (2007)
Petrović, N.Z., Bohra, M.: General Jacobi elliptic function expansion method applied to the generalized (3+1)-dimensional nonlinear Schödinger equation. Opt. Quant. Electron. (2016). https://doi.org/10.1007/s11082-016-0522-1
Rodríguez, C.M.: Orbits in General Relativity: The Jacobian Elliptic Functions. Il Nuovo Cimento 98B, 87–96 (1987)
Siegel, C.L.: Topics in Complex Function Theory. Volume I: Elliptic Functions and Uniformization Theory. Wiley, New York (1988)
Zudilin, V.V.: Thetanulls and differential equations. Sbornik Mathematics 191, 1–45 (2000)
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Campos, R.G. (2018). A Fractional Partial Differential Equation for Theta Functions. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_26
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DOI: https://doi.org/10.1007/978-3-319-96827-8_26
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