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Mathematica > Mathematics and Algorithms > Mathematical Functions > Special Functions > Elliptic Functions > SiegelTheta >

BUILT-IN MATHEMATICA SYMBOL

SiegelTheta

gives the Siegel theta function

with Riemann modular matrix

and vector s.

SiegelTheta

gives the Siegel theta function

with characteristics

and

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

The matrix must be symmetric, with positive definite imaginary part.

If is a matrix, the vectors s and v or 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 (2)

Evaluate numerically:

In[1]:=

Out[1]=

In[1]:=

Out[1]=

Scope (4)

Evaluate SiegelTheta for higher-dimensional arguments:

Evaluate for complex arguments:

Evaluate to high precision:

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

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)