SiegelTheta

SiegelTheta[Ω,s]

gives the Siegel theta function with Riemann modular matrix Ω and vector s.

SiegelTheta[{ν1,ν2},Ω,s]

gives the Siegel theta function with characteristics ν1 and ν2.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • The matrix Ω must be symmetric, with positive definite imaginary part.
  • If Ω is a p×p matrix, the vectors s and v or νi must have length p.
  • , where n ranges over all possible vectors in the p-dimensional integer lattice.
  • , where n ranges over all possible vectors in the p-dimensional integer lattice.
  • SiegelTheta can be evaluated to arbitrary numerical precision.

Examples

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Basic Examples  (3)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Scope  (7)

Numerical Evaluation  (3)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate efficiently at high precision:

Specific Values  (2)

Value at zero:

Find a value of x for which Re[SiegelTheta[{{I,1},{1, I}},{x,x/3}]]=1:

Visualization  (2)

Plot the SiegelTheta function for various parameters:

Plot the real part of SiegelTheta[(ⅈ 1; 1 ⅈ),{x+i y,(x+i y )/3}]:

Plot the imaginary part of SiegelTheta[(ⅈ 1; 1 ⅈ),{x+i y,(x+i y )/3}]:

Generalizations & Extensions  (2)

SiegelTheta with characteristics and :

SiegelTheta with characteristics simplifies symbolically for special arguments:

Applications  (2)

Plot of the absolute value of SiegelTheta in the complex plane:

Define an Abelian function:

Plot of the real part:

Properties & Relations  (2)

In one dimension, SiegelTheta coincides with the EllipticTheta functions:

SiegelTheta satisfies the equations:

Possible Issues  (2)

SiegelTheta requires a symmetric matrix:

The symmetric part:

Machine precision may be insufficient to obtain a correct answer:

Use arbitrary precision to check the result:

Neat Examples  (1)

Introduced in 2007
 (6.0)