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Jacobi theta functions and related functions
===============================================================================



The theta functions are functions of two variables:

* `z` is the *argument*, an arbitrary real or complex number

* `q` is the *nome*, which must be a real or complex number
  in the unit disk (i.e. `|q| < 1`). For `|q| \ll 1`, the
  series converge very quickly, so the Jacobi theta functions
  can efficiently be evaluated to high precision.

The compact notations `\theta_n(q) = \theta_n(0,q)`
and `\theta_n = \theta_n(0,q)` are also frequently
encountered. Finally, Jacobi theta functions are frequently
considered as functions of the half-period ratio `\tau`
and then usually denoted by `\theta_n(z|\tau)`.




Jacobi theta function `\theta_1(z, q)`
-------------------------------------------------------------------------------

.. method:: ctx.jacobi_theta1(x, q)

    where ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflint``.

    Returns the Jacobi theta function `\displaystyle  \theta_1(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty} (-1)^n q^{n^2+n\,} \sin((2n+1)z)` 

    Note: the Amath version needs to be adapted.



|JacobiTheta1|

.. |JacobiTheta1| image:: ../_static/ExplicitSurfaces/RealFunctions/JacobiTheta1.3D.xml.jpg
   :width: 30 %



**Left figure**: real ("silver") and imaginary ("gold") part of the Jacobi theta function `\theta_1(z, q)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.





    See also Wikipedia :cite:p:`WikipediaFun170`, MathWorld :cite:p:`WolframFun170`, MathWorld :cite:p:`WolframFun170a`, NIST :cite:p:`DLMFun170`, :cite:t:`Ehrhardt2018` (3.2.13), Flint :cite:p:`FlintFun170`.


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.JacobiTheta(1, 0.8, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.JacobiTheta(2, 0.8, '0.51')
        xreal('5.3518479027559984754E-1')



    An example with real input for `\theta_1(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '3.8'; q = '0.7'
        >>> \mathrm{d}x = dec.jtheta(1, x, q); mx = mpm.jtheta(1, x, q); gx = gmp.jtheta(1, x, q)
        >>> fx = fpm.jtheta(1, x, q); ax = apm.jtheta(1, x, q)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  -2.876223175938890139154258006053690969477E-1
        mpm:  -2.876223175938890139154258006053690969477e-1
        gmp:  -2.876223175938890139154258006053690969477E-01
        fpm:  -2.87622317593889E-01
        apm:  -2.876223175938890139154258006053690969480e-1 (-2.326e-35%)



    An example with complex input for `\theta_1(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '11.0 + 3.0j'; q = '0.7 + 0.1j'
        >>> \mathrm{d}z = dec.jtheta(1, z, q); mz = mpm.jtheta(1, z, q); gz = gmp.jtheta(1, z, q)
        >>> fz = fpm.jtheta(1, z, q); az = apm.jtheta(1, z, q)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: 1.3257637053506044723E+10              - 2.5508683032905869130E+9j
        mpm: 1.3257637053506044723e+10              - 2.5508683032905869130e+9j
        gmp: 1.3257637053506044723E+10              - 2.5508683032905869130E+09j
        fpm: 1.32576370535060E+10                   - 2.55086830329060E+09j
        apm: 1.3257637053506044630e+10 (4.315e-14%) - 2.5508683032905869160e+9 (-2.258e-13%)j



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Jacobi theta function `\theta_2(z, q)`
-------------------------------------------------------------------------------

.. method:: ctx.jacobi_theta2(x, q)

    where ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflint``.

    Returns the Jacobi theta function `\displaystyle  \theta_2(z,q) = 2 q^{1/4} \sum_{n=0}^{\infty} q^{n^{2\,} + n} \cos((2n+1)z)` 


    Note: the Amath version needs to be adapted.



|JacobiTheta2|

.. |JacobiTheta2| image:: ../_static/ExplicitSurfaces/RealFunctions/JacobiTheta2.3D.xml.jpg
   :width: 30 %



**Left figure**: real ("silver") and imaginary ("gold") part of the Jacobi theta function `\theta_2(z, q)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.





    See also Wikipedia :cite:p:`WikipediaFun170`, MathWorld :cite:p:`WolframFun170`, MathWorld :cite:p:`WolframFun170a`, NIST :cite:p:`DLMFun170`, :cite:t:`Ehrhardt2018` (3.2.13), Flint :cite:p:`FlintFun170`.


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.JacobiTheta(1, 0.8, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.JacobiTheta(2, 0.8, '0.51')
        xreal('5.3518479027559984754E-1')


    An example with real input for `\theta_2(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '3.8'; q = '0.7'
        >>> \mathrm{d}x = dec.jtheta(2, x, q); mx = mpm.jtheta(2, x, q); gx = gmp.jtheta(2, x, q)
        >>> fx = fpm.jtheta(2, x, q); ax = apm.jtheta(2, x, q)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  -8.802379983271747955434299445752594240170E-1
        mpm:  -8.802379983271747955434299445752594240170e-1
        gmp:  -8.802379983271747955434299445752594240170E-01
        fpm:  -8.80237998327175E-01
        apm:  -8.802379983271747955434299445752594240180e-1 (-1.375e-35%)



    An example with complex input for `\theta_2(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '11.0 + 3.0j'; q = '0.7 + 0.1j'
        >>> \mathrm{d}z = dec.jtheta(2, z, q); mz = mpm.jtheta(2, z, q); gz = gmp.jtheta(2, z, q)
        >>> fz = fpm.jtheta(2, z, q); az = apm.jtheta(2, z, q)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: 2.6679401530814916188E+11              - 3.0867614993820689042E+11j
        mpm: 2.6679401530814916188e+11              - 3.0867614993820689042e+11j
        gmp: 2.6679401530814916188E+11              - 3.0867614993820689042E+11j
        fpm: 2.66794015308149E+11                   - 3.08676149938205E+11j
        apm: 2.6679401530814916130e+11 (4.123e-14%) - 3.0867614993820688890e+11 (-3.758e-14%)j






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Jacobi theta function `\theta_3(z, q)`
-------------------------------------------------------------------------------

.. method:: ctx.jacobi_theta3(x, q)

    where ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflint``.


    Returns the Jacobi theta function `\displaystyle  \theta_3(z,q) = 1 + 2 \sum_{n=1}^{\infty} q^{n^2\,} \cos(2 n z)` 



    Note: the Amath version needs to be adapted.



|JacobiTheta3|

.. |JacobiTheta3| image:: ../_static/ExplicitSurfaces/RealFunctions/JacobiTheta3.3D.xml.jpg
   :width: 30 %



**Left figure**: real ("silver") and imaginary ("gold") part of the Jacobi theta function `\theta_3(z, q)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.





    See also Wikipedia :cite:p:`WikipediaFun170`, MathWorld :cite:p:`WolframFun170`, MathWorld :cite:p:`WolframFun170a`, NIST :cite:p:`DLMFun170`, :cite:t:`Ehrhardt2018` (3.2.13), Flint :cite:p:`FlintFun170`.


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.JacobiTheta(1, 0.8, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.JacobiTheta(2, 0.8, '0.51')
        xreal('5.3518479027559984754E-1')



    An example with real input for `\theta_3(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '3.8'; q = '0.7'
        >>> \mathrm{d}x = dec.jtheta(3, x, q); mx = mpm.jtheta(3, x, q); gx = gmp.jtheta(3, x, q)
        >>> fx = fpm.jtheta(3, x, q); ax = apm.jtheta(3, x, q)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  8.802381825612654435491537366191521291923E-1
        mpm:  8.802381825612654435491537366191521291923e-1
        gmp:  8.802381825612654435491537366191521291923E-01
        fpm:  8.80238182561266E-01
        apm:  8.802381825612654435491537366191521291940e-1 (1.375e-35%)




    An example with complex input for `\theta_3(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '11.0 + 3.0j'; q = '0.7 + 0.1j'
        >>> \mathrm{d}z = dec.jtheta(3, z, q); mz = mpm.jtheta(3, z, q); gz = gmp.jtheta(3, z, q)
        >>> fz = fpm.jtheta(3, z, q); az = apm.jtheta(3, z, q)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: -2.6679401441538351941E+11               + 3.0867614546580330581E+11j
        mpm: -2.6679401441538351941e+11               + 3.0867614546580330581e+11j
        gmp: -2.6679401441538351941E+11               + 3.0867614546580330581E+11j
        fpm: -2.66794014415383E+11                    + 3.08676145465801E+11j
        apm: -2.6679401441538351880e+11 (-4.086e-14%) + 3.0867614546580330430e+11 (3.758e-14%)j






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Jacobi theta function `\theta_4(z, q)`
-------------------------------------------------------------------------------

.. method:: ctx.jacobi_theta4(x, q)

    where ``ctx`` is ``math53``, ``ctxboost`` or ``ctxflint``.

    Returns the Jacobi theta function `\displaystyle  \theta_4(z,q) = 1 + 2 \sum_{n=1}^{\infty} (-q)^{n^2\,} \cos(2 n z)` 


    Note: the Amath version needs to be adapted.



|JacobiTheta4|

.. |JacobiTheta4| image:: ../_static/ExplicitSurfaces/RealFunctions/JacobiTheta4.3D.xml.jpg
   :width: 30 %



**Left figure**: real ("silver") and imaginary ("gold") part of the Jacobi theta function `\theta_4(z, q)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.





    See also Wikipedia :cite:p:`WikipediaFun170`, MathWorld :cite:p:`WolframFun170`, MathWorld :cite:p:`WolframFun170a`, NIST :cite:p:`DLMFun170`, :cite:t:`Ehrhardt2018` (3.2.13), Flint :cite:p:`FlintFun170`.


    An example in Python

    .. code-block:: pycon

        >>> from xlcalcnet import xreal
        >>> xreal.JacobiTheta(1, 0.8, 0.5)
        xreal('5.2359877559829887307E-1')
        >>> xreal.JacobiTheta(2, 0.8, '0.51')
        xreal('5.3518479027559984754E-1')



    An example with real input for `\theta_4(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; x = '3.8'; q = '0.7'
        >>> \mathrm{d}x = dec.jtheta(4, x, q); mx = mpm.jtheta(4, x, q); gx = gmp.jtheta(4, x, q)
        >>> fx = fpm.jtheta(4, x, q); ax = apm.jtheta(4, x, q)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  2.876275982849924933190081281315277893390E-1
        mpm:  2.876275982849924933190081281315277893390e-1
        gmp:  2.876275982849924933190081281315277893390E-01
        fpm:  2.87627598284992E-01
        apm:  2.876275982849924933190081281315277893400e-1 (2.587e-35%)



    An example with complex input for `\theta_4(z,q)`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '11.0 + 3.0j'; q = '0.7 + 0.1j'
        >>> \mathrm{d}z = dec.jtheta(4, z, q); mz = mpm.jtheta(4, z, q); gz = gmp.jtheta(4, z, q)
        >>> fz = fpm.jtheta(4, z, q); az = apm.jtheta(4, z, q)
        >>> mpm.show([\mathrm{d}z, mz, gz, fz, az], aligned=True)
        dec: -1.3378357586306498825E+10               + 2.5798286002171674621E+9j
        mpm: -1.3378357586306498825e+10               + 2.5798286002171674621e+9j
        gmp: -1.3378357586306498825E+10               + 2.5798286002171674621E+09j
        fpm: -1.33783575863064E+10                    + 2.57982860021718E+09j
        apm: -1.3378357586306498730e+10 (-4.313e-14%) + 2.5798286002171674650e+9 (2.252e-13%)j