: Riemann theta function and
: a Riemann matrix
: Riemann theta function and
: a Riemann matrix
when 
Thus
is periodic, with period 1, in
each element of 
. More generally,
: Riemann theta function,
: base of exponential function and
: a Riemann matrix
with 
, 
. This is the
quasi-periodicity property of the Riemann theta function. It determines
the Riemann theta function up to a constant factor. The set of points
form a 
-dimensional lattice,
the period lattice of the Riemann theta function.
Again, with ,
![\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}+\mathbf{m}_{1}}{%
\boldsymbol{{\beta}}+\mathbf{m}_{2}}\/}\nolimits\!\left(\mathbf{z}\middle|%
\boldsymbol{{\Omega}}\right)=e^{{2\pi i\boldsymbol{{\alpha}}\cdot\mathbf{m}_{2%
}}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{%
{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).](./21/3/E4.png){kind=small}
![\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{%
\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)](./21/3/m14.png){kind=small}
: Riemann theta function with characteristics,
: base of exponential function,
: a Riemann matrix,
: -dimensional vector and
: -dimensional vector
on the right-hand-side was given
incorrectly as .
Reported 2012-08-27 by Klaas Vantournhout
Because of this property, the elements of and
are usually restricted to 
, without loss of generality.
![\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{%
\beta}}}\/}\nolimits\!\left(\mathbf{z}+\mathbf{m}_{1}+\boldsymbol{{\Omega}}%
\mathbf{m}_{2}\middle|\boldsymbol{{\Omega}}\right)=e^{{2\pi i\left(\boldsymbol%
{{\alpha}}\cdot\mathbf{m}_{1}-\boldsymbol{{\beta}}\cdot\mathbf{m}_{2}-\frac{1}%
{2}\mathbf{m}_{2}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{m}_{2}-\mathbf{m}_{2}%
\cdot\mathbf{z}\right)}}\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{%
\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!\left(\mathbf{z}\middle|%
\boldsymbol{{\Omega}}\right).](./21/3/E5.png){kind=small}
: Riemann theta function with characteristics,
: base of exponential function,
: a Riemann matrix,
: -dimensional vector and
: -dimensional vector
For Riemann theta functions with half-period characteristics,
![\mathop{\theta\!\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{%
\beta}}}\/}\nolimits\!\left(-\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=(-%
1)^{{4\boldsymbol{{\alpha}}\cdot\boldsymbol{{\beta}}}}\mathop{\theta\!\genfrac%
{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\/}\nolimits\!%
\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right).](./21/3/E6.png){kind=small}
: Riemann theta function with characteristics,
: a Riemann matrix,
: -dimensional vector and
: -dimensional vector
See also §20.2(iii) for the case 
and classical theta
functions.
