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NumberTheory`SiegelTheta`
The Siegel theta function 
is defined as
where  is a  symmetric complex matrix  with imaginary part  positive definite,  is a complex vector of dimension  , and  is an integer vector of dimension  that ranges over the entire 
-dimensional lattice of integers. This function was initially investigated by Riemann and Weierstrass and further studies were done by Frobenius and Poincaré. These investigations represent some of the most significant accomplishments of nineteenth century mathematics.
The Siegel theta function.
This loads the package.
In[1]:= << NumberTheory`SiegelTheta`
This evaluates 
for particular arguments in 2-space.
In[2]:= SiegelTheta[{{1+I,2+I}, {2+I,-1+4I}}, {1.2, 2.3+.3I}]
Out[2]= 
This is a slow, brute force way to get the same answer. (The tails beyond 
10 and 10 die off quickly.)
In[3]:= Sum[E^(Pi I {t1,t2}.{{1+I,2+I}, {2+I,-1+4I}}.{t1,t2} +
2 Pi I {t1,t2} . {1.2, 2.3+.3I}),
{t1, -10, 10}, {t2, -10, 10}]
Out[3]= 
If the imaginary part of  is not positive definite, 
is not defined.
In[4]:= SiegelTheta[{{1+I,2+I}, {2+I,-1-4I}}, {1.2, 2.3+.3I}]
SiegelTheta::npd: The imaginary part of the matrix {{1 + I, 2 + I}, {2 + I, -1 - 4 I}} is not positive definite.
Out[4]= 
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT. SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION. | |
