The Riemann theta function is a complex function of complex variables that occurs in the construction
of quasi-periodic solutions of various equations in mathematical physics (Deconinck
et al. 2004). Any Abelian function can
be expressed as a ratio of homogeneous polynomials of the Riemann theta function
(Igusa 1972, Deconinck et al. 2004).
Riemann (1857) first considered these functions associated with Riemann surfaces, and the most general form of the Riemann theta function defined above
was first considered by Wirtinger (1895).
An overview of the properties of the Riemann theta function is given by Mumford (1983, 1984, 1991), and algorithms for numeric computations have been developed by Deconinck
et al. (2004).
Belokolos; E. D.; Bobenko; A. I.; Enol'skii; V. Z.; Its; A. R.; and Matveev; V. B. Algebro-Geometric Approach to Nonlinear
Integrable Problems. Berlin: Springer-Verlag, 1994.Deconinck, B.;
Heil, M.; Bobenko, A.; van Hoeij, M.; and Schmies, M. "Computing Riemann Theta
Functions." Math. Comput.73, 1417-1442, 2004.Dubrovin;
B. A. "Theta Functions and Nonlinear Equations." Russian Math.
Surveys36, 11-80, 1981.Igusa, J.-I. Theta
Functions. New York: Springer-Verlag, 1972.Itô, K. (Ed.).
"Abelian Integrals." §3.L in Encyclopedic
Dictionary of Mathematics, 2nd ed., Vol. 1. Cambridge, MA: MIT Press,
p. 9, 1987.Jacobi; C. G. J. Fundamenta Nova Theoriae
Functionum Ellipticarum. Königsberg, Germany, 1829.Mumford,
D. Tata
Lectures on Theta. I. Boston, MA: Birkhäuser, 1983.Mumford,
D. Tata
Lectures on Theta. II. Jacobian Theta Functions and Differential Equations.
Boston, MA: Birkhäuser, 1984.Mumford, D. Tata
Lectures on Theta. III. Boston, MA: Birkhäuser, 1991.Riemann,
G. F. B. "Theorie der Abel'schen Functionen." J. reine angew.
Math.54, 101-155, 1857.Wirtinger, W. Untersuchungen über
Thetafunctionen. Leipzig, Germany: Teubner, 1895.
complex variables that occurs in the construction
of quasi-periodic solutions of various equations in mathematical physics (Deconinck
et al. 2004). Any Abelian function can
be expressed as a ratio of homogeneous polynomials of the Riemann theta function
(Igusa 1972, Deconinck et al. 2004).
matrix 
be positive definite,
and 
be a row vector with coefficients in 
. Then the Riemann theta function is defined by
![theta(u|F)=sum_(m)exp[2pii(m^(T)u+1/2mF^(T)m)].](/images/equations/RiemannThetaFunction/NumberedEquation1.svg)
