Riemann Theta Function
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).
Let the imaginary part of a 
matrix 
be positive definite, and 
be a row vector with coefficients in 
. Then the Riemann theta function is defined by
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).
REFERENCES:
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. Surveys 36, 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.
