Elliptic Functions, Theta Functions, and Riemann Surfaces

1974 ◽  
Vol 28 (127) ◽  
pp. 875
Author(s):  
Y. L. L. ◽  
Harry E. Rauch ◽  
Aaron Lebowitz
Keyword(s):  
Riemann Surfaces ◽  
Theta Functions ◽  
Elliptic Functions

Theta functions and elliptic functions

2016 ◽  
pp. 384-403
Author(s):  
J. Hietarinta ◽  
N. Joshi ◽  
F. W. Nijhoff
Keyword(s):  
Theta Functions ◽  
Elliptic Functions

Determinantal representations of elliptic curves via Weierstrass elliptic functions

2018 ◽  
Vol 34 ◽  
pp. 125-136 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Elliptic Curves ◽  
Algebraic Curve ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Determinantal Representation ◽  
Ternary Form ◽  
Riemann Theta Functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.

LEVEL-ONE SU(N) WZNW MODELS ON HIGHER-GENUS RIEMANN SURFACES

1993 ◽  
Vol 08 (17) ◽  
pp. 2955-2972 ◽  
Author(s):  
M. ALIMOHAMMADI ◽  
H. ARFAEI
Keyword(s):  
Partition Function ◽  
Simple Form ◽  
Riemann Surfaces ◽  
Theta Functions ◽  
Modular Invariance ◽  
Fusion Rules ◽  
Group Manifold ◽  
Special Cases ◽  
Wznw Model ◽  
Higher Genus

Using factorization properties and fusion rules, we find the higher-genus partition function and two-point correlators for the SU (N)1 WZNW model. The result has simple form in terms of higher-genus theta functions on the group manifold. The previously known results of SU (2)1 and SU (3)1 are also obtained as special cases. This method, combined with other considerations such as modular invariance, can be extended to the nonsimply laced groups and higher-level WZNW models.

scholarly journals Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ernest X. W. Xia ◽  
Olivia X. M. Yao
Keyword(s):  
Eisenstein Series ◽  
Theta Functions ◽  
Elliptic Functions ◽  
The Other ◽  
Fundamental Theory ◽  
Convolution Sums

Based on the theories of Ramanujan's elliptic functions and the (p,k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

scholarly journals VIII. A memoir on internal functions

1902 ◽  
Vol 199 (312-320) ◽  
pp. 411-500 ◽  
Keyword(s):  
Theta Functions ◽  
Elliptic Functions ◽  
Single Variable ◽  
Hyperbolic Functions ◽  
Gamma Functions ◽  
Physical Problems ◽  
Simple Class ◽  
Transcendental Functions ◽  
Natural Groups ◽  
Modern Mathematics

1. Since the fundamental discoveries of Weierstrass, much progress has been made with regard to uniform transcendental functions; but the advances of modern mathematics appear to have included no attempt formally to classify and investigate the properties of natural groups of such functions. Consider, for instance, the case of transcendental integral functions which admit one possible essential singularity at infinity. They form the most simple class of uniform functions of a single variable, and yet of them we know, broadly speaking, the nature of but four types:- (1) The exponential function, with which are associated circular and (rectangular) hyperbolic functions; (2) The gamma functions; (3) The elliptic functions and functions derived therefrom, such as the theta functions and Appell’s generalisation of the Eulerian functions; (4) Certain functions which arise in physical problems (such as x -s J„ ( x )) whose properties have been extensively investigated for physical purposes.

scholarly journals XXI. A Memoir on the Single and Double Theta-Functions

1880 ◽  
Vol 171 ◽  
pp. 897-1002
Keyword(s):  
Exponential Function ◽  
Dimensional Space ◽  
Theta Functions ◽  
Plane Curve ◽  
Elliptic Functions ◽  
Square Root ◽  
Ordinary Space ◽  
Abelian Functions ◽  
Series Of Exponentials ◽  
Irrational Function

The Theta-Functions, although arising historically from the Elliptic Functions, may be considered as in order of simplicity preceding these, and connecting themselves directly with the exponential function (e x or) exp. x ; viz., they may be defined each of them as a sum of a series of exponentials, singly infinite in the case of the single functions, doubly infinite in the case of the double functions ; and so on. The number of the single functions is = 4; and the quotients of these, or say three of them each divided by the fourth, are the elliptic functions sn, cn, d n ; the number of the double functions is (4 2 = ) 16 ; and the quotients of these, or say fifteen of them each divided by the sixteenth, are the hyper-elliptic functions of two arguments depending on the square root of a sex tic function : generally the number of the p -tuple theta-functions is = 4 p ; and the quotients of these, or say all but one of them each divided by the remaining function, are the Abelian functions of p arguments depending on the irrational function y defined by the equation F ( x, y ) = 0 of a curve of deficiency p ). If instead of connecting the ratios of the functions with a plane curve we consider the functions themselves as coordinates of a point in a (4 p —1)dimensional space, then we have the single functions as the four coordinates of a point on a quadri-quadric curve (one-fold locus) in ordinary space; and the double functions as the sixteen coordinates of a point on a quadri-quadric two-fold locus in 15-dimensional space, the deficiency of this two-fold locus being of course = 2.

Sign in / Sign up

Export Citation Format

Share Document