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Weierstrass elliptic functions, in terms of (real) lattice invariants `g_2, g_3`
=====================================================================================

The Weierstrass functions take real values on the real axis iff the lattice is fixed under complex conjugation, or, equivalently, when `g_2, g_3 \in \mathbb{R}`.

See also: https://gist.github.com/stla/d771e0a8c351d16d186c79bc838b6c48




.. _rst_mpm_wpg: 

Weierstrass function `\wp_g(z, g_2, g_3)` (also DAMath)
-------------------------------------------------------------------------------


.. method:: ctxflint.weierstrass_p_g(z, g2, g3)


    Computes Weierstrass's elliptic function `\wp_g(z; g_2, g_3)`. 

    We have `\wp_g(tz; t^{-4} g_2, t^{-6} g_3) = t^{-2} \wp_g(z; g_2, g_3)` and `\wp_g(i z; g_2, g_3) = -\wp_g(z; g_2, -g_3)`.


    See also MathWorld :cite:p:`WolframFun190`, Flint :cite:p:`FlintFun190`.


    Returns the Weierstrass function `\wp_g(x, g_2, g_3)` based on the lattice invariants `g_2`, `g_3`.


    In AMath the computation of the Weierstass elliptic function (for real and imaginary
    arguments) is based on the lattice invariants `g_2`, `g_3` or the lattice roots `e_1, e_2, e_3 = -e_1-e_2`, if the discriminant of the cubic equation `4x^3 - g_2x - g_3 = 0` is positive

    .. math:: \Delta = g_2^3 - 27g_3^2 = 16(e_2-e_3)^2 (e_3-e_1)^2 (e_1-e_2)^2;

    symbolically written as `\wp_g(x, g_2, g_3)` or `\wp_e(x,e_1,e_2)`. See also 



    When `g_2=1, g_3=0` the result is  `\wp_l(x)` and for `g_2=g_3=0` it is `x-2`. If `g_2^2 - 27g_3^2>0` all lattice roots are real, and `\wp'_e(x,e_1,e_2)` is returned, otherwise the following Jacobi relation is used:

    .. math:: \wp_g(x, g_2, g_3) = e_2 + H \frac{1+\mathrm{cn}(u,k)}{1-\mathrm{cn}(u,k)} , \quad k = \sqrt{\tfrac{1}{2}\frac{3e_2}{4H}}, \quad u = 2x \sqrt{H}, \quad H = \sqrt{(e_2-e_1)(e_2-e_3)}.

    See also Wikipedia :cite:p:`WikipediaFun195a`, MathWorld :cite:p:`WolframFun190`, :cite:t:`Ehrhardt2018` (3.2.17.5).



    Returns the Weierstrass function `\wp_g(iy,g_2, g_3) = -\wp_g(y,g_2, -g_3)`. See also: MathWorld :cite:p:`WolframFun191`, :cite:t:`Ehrhardt2018` (3.2.17.7).




|11a_TestWeierstrassP_re| `\quad` |11b_TestWeierstrassP_im| `\quad` |11c_TestWeierstrassP_abs|

.. |11a_TestWeierstrassP_re| image:: ../_static/ExplicitSurfaces/CplxElliptic/11a_TestWeierstrassP_re.3D.xml.jpg
   :width: 30 %

.. |11b_TestWeierstrassP_im| image:: ../_static/ExplicitSurfaces/CplxElliptic/11b_TestWeierstrassP_im.3D.xml.jpg
   :width: 30 %

.. |11c_TestWeierstrassP_abs| image:: ../_static/ExplicitSurfaces/CplxElliptic/11c_TestWeierstrassP_abs.3D.xml.jpg
   :width: 30 %



**Left figure**: real part of the Weierstrass function `\wp_g(z, g_2, g_3)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Middle figure**: imaginary part of the Weierstrass function `\wp_g(z, g_2, g_3)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Right figure**:  absolute value of the Weierstrass function `\wp_g(z, g_2, g_3)`, with color-coded phase. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.






    For `\Delta = g_2^3 - 27 g_3^2 \ne 0`, we first compute the elliptic half periods `\omega_1, \omega_2` from the lattice  invariants `g_2, g_3`  (see :ref:`EllipticHalfPeriodsG <rst_mpm_elliptic_halfperiods_from_invariants>`), and set `\omega = \omega_1, \tau = \omega_2/\omega_1`. 
    Then `\wp_g(z; g_2, g_3) = \wp(z| \omega, \tau)` (see :ref:`WeierstrassP <rst_mpm_wpg_by_tau>`).


    For `\Delta = 0, g_2 > 0, g_3 \ne 0`, the function `\wp_g(\cdot)` degenerates to a simply periodic function, which can be expressed in closed form in terms of elementary functions:

    For `\displaystyle \Delta = 0, g_2 > 0, g_3 < 0: \quad  \wp_g(z; g_2, g_3) = c + \frac{3c}{\sinh^2\left(\sqrt{3c} z \right)}, \quad \text{where } c = \sqrt{g_2/12}`.

    For `\displaystyle \Delta = 0, g_2 > 0, g_3 > 0: \quad  \wp_g(z; g_2, g_3) = c + \frac{3c}{\sin^2\left( \sqrt{3c}z \right)}, \quad \text{where } c = \sqrt{g_2/12}`.


    If both `g_2` and `g_3` are zero, the function `\wp_g(\cdot)` degenerates to a function that is not periodic at all, namely `\wp_g(z; 0, 0) = z^{-2}`.






    An example with real input for `z` and with purely imaginary input for `\tau`, producing real output:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; z = '0.1'; tau = '0.0 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_p(z, tau); mx = mpm.weierstrass_p(z, tau)
        >>> gx = gmp.weierstrass_p(z, tau)
        >>> fx = fpm.weierstrass_p(z, tau); ax = apm.weierstrass_p(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.001202913728593351620333736461324760825E+2
        mpm:  1.001202913728593351620333736461324760825e+2
        gmp:  1.001202913728593351620333736461324760825E+02
        fpm:  1.00120291372859E+02
        apm:  1.001202913728593351620333736461324760825e+2 (1.321e-38%)




    An example with complex input for `z` and `\tau`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '0.1 + 0.5j'; tau = '0.7 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_p(z, tau); mx = mpm.weierstrass_p(z, tau)
        >>> gx = gmp.weierstrass_p(z, tau)
        >>> fx = fpm.weierstrass_p(z, tau); ax = apm.weierstrass_p(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax], aligned=True)
        dec:  -2.3859027427781674104E+0
        mpm:  -2.3859027427781674104e+0
        gmp:  -2.3859027427781674104E+00
        fpm:  -2.38590274277817E+00
        apm:  (-2.3859027427781674104e+0 (-2.038e-17%) + 2.4323666240267878020e-2 (1.982e-15%)j)



    
|newpage|


.. _rst_mpm_wpg_prime: 

Weierstrass function, first derivative: `\wp_g'(z, g_2, g_3)`
-------------------------------------------------------------------------------


.. method:: ctxflint.weierstrass_p_prime_(z, g2, g3)

    Computes the first derivative of the Weierstrass function, `\wp_g'(z, g_2, g_3)`. 

    We have `\wp_g'(tz; t^{-4} g_2, t^{-6} g_3) = t^{-3} \wp_g'(z; g_2, g_3)` and `\wp_g'(i z; g_2, g_3) = i\wp_g'(z; g_2, -g_3)`. 


    See also MathWorld :cite:p:`WolframFun191`, Flint :cite:p:`FlintFun190`.



    Returns the Weierstrass function `\wp_g(x, g_2, g_3)` based on the lattice invariants `g_2`, `g_3`.

    .. math:: \wp'_g(x, g_2, g_3) = -H^{3/2} \frac{\mathrm{cn}(u,k)\mathrm{dn}(u,k)}{1-\mathrm{cn}(u,k)} , \quad k = \sqrt{\tfrac{1}{2}\frac{3e_2}{4H}}, \quad u = 2x \sqrt{H}, \quad H = \sqrt{(e_2-e_1)(e_2-e_3)}.


    See also: MathWorld :cite:p:`WolframFun191`, :cite:t:`Ehrhardt2018` (3.2.17.6).



|12a_TestWeierstrassPPrime_re| `\quad` |12b_TestWeierstrassPPrime_im| `\quad` |12c_TestWeierstrassPPrime_abs|

.. |12a_TestWeierstrassPPrime_re| image:: ../_static/ExplicitSurfaces/CplxElliptic/12a_TestWeierstrassPPrime_re.3D.xml.jpg
   :width: 30 %

.. |12b_TestWeierstrassPPrime_im| image:: ../_static/ExplicitSurfaces/CplxElliptic/12b_TestWeierstrassPPrime_im.3D.xml.jpg
   :width: 30 %

.. |12c_TestWeierstrassPPrime_abs| image:: ../_static/ExplicitSurfaces/CplxElliptic/12c_TestWeierstrassPPrime_abs.3D.xml.jpg
   :width: 30 %



**Left figure**: real part of the Weierstrass function, first derivative: `\wp_g'(z, g_2, g_3)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Middle figure**: imaginary part of the Weierstrass function, first derivative: `\wp_g'(z, g_2, g_3)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Right figure**:  absolute value of the Weierstrass function, first derivative: `\wp_g'(z, g_2, g_3)`, with color-coded phase. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.






    For `\Delta = g_2^3 - 27 g_3^2 \ne 0`, we first compute the elliptic half periods `\omega_1, \omega_2` from the lattice  invariants `g_2, g_3` (see :ref:`EllipticHalfPeriodsG <rst_mpm_elliptic_halfperiods_from_invariants>`), and set `\omega = \omega_1, \tau = \omega_2/\omega_1`. Then `\wp_g'(z; g_2, g_3) = \wp'(z| \omega, \tau)` (see :ref:`WeierstrassPPrime <rst_mpm_wpg_prime_by_tau>`).


    For `\Delta = 0, g_2 > 0, g_3 \ne 0`, the function `\wp_g'(\cdot)` degenerates to a simply periodic function, which can be expressed in closed form in terms of elementary functions:


    For `\Delta = 0, g_2 > 0, g_3 < 0: \quad  \displaystyle \wp_g'(z; g_2, g_3) = -6 \sqrt{3} c^{3/2} \mathrm{coth}(\sqrt{3c}z) \mathrm{csch}^2(\sqrt{3c}z),  \quad  \text{where } c = \sqrt{g_2/12}`.


    For `\Delta = 0, g_2 > 0, g_3 > 0: \quad  \displaystyle \wp_g'(z; g_2, g_3) = -6 \sqrt{3} c^{3/2} \mathrm{cot}(\sqrt{3c}z) \mathrm{csc}^2(\sqrt{3c}z),  \quad  \text{where } c = \sqrt{g_2/12}`.



    If both `g_2` and `g_3` are zero, the function `\wp_g'(\cdot)` degenerates to a function that is not periodic at all, namely  `\wp_g'(z; 0, 0) = -2 z^{-3}`.







    An example with real input for `z` and with purely imaginary input for `\tau`, producing real output:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; z = '0.1'; tau = '0.0 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_p_prime(z, tau); mx = mpm.weierstrass_p_prime(z, tau)
        >>> gx = gmp.weierstrass_p_prime(z, tau)
        >>> fx = fpm.weierstrass_p_prime(z, tau); ax = apm.weierstrass_p_prime(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  -1.997612036772329460780279347633189585828E+3
        mpm:  -1.997612036772329460780279347633189585828e+3
        gmp:  -1.997612036772329460780279347633189585828E+03
        fpm:  -1.99761203677233E+03
        apm:  -1.997612036772329460780279347633189585828e+3 (-1.883e-38%)




    An example with complex input for `z` and `\tau`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '0.1 + 0.5j'; tau = '0.7 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_p_prime(z, tau); mx = mpm.weierstrass_p_prime(z, tau)
        >>> gx = gmp.weierstrass_p_prime(z, tau)
        >>> fx = fpm.weierstrass_p_prime(z, tau); ax = apm.weierstrass_p_prime(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax], aligned=True)
        dec: 1.4787544521497118400E+1
        mpm: 1.4787544521497118400e+1
        gmp: 1.4787544521497118400E+01
        fpm: 1.47875445214971E+01
        apm: 1.4787544521497118399e+1 (5.664e-17%) - 2.3205214399360215619e+1 (-3.609e-17%)j




|newpage|

.. _rst_mpm_wpg_inv: 

Inverse Weierstrass function `\wp_g^{-1}(z, g_2, g_3)`
-------------------------------------------------------------------------------


.. method:: ctxflint.weierstrass_p_invG(z, g2, g3)


    Computes the Inverse Weierstrass function `\wp_g^{-1}(z, g_2, g_3)`

    See also MathWorld :cite:p:`WolframFun192`, Flint :cite:p:`FlintFun190`.


    .. math :: \wp_g^{-1}(z; g_2, g_3) = \wp^{-1}(z| \tau = \omega_2/\omega_1, \omega = \omega_1)


    Returns the functional inverse `\wp^{-1}_g` of the Weierstrass function for `y \ge e_1`, i.e. the smallest positive `x` with `\wp^{-1}_g(y,g_2,g_3) = y`, if it exists. See also: MathWorld :cite:p:`WolframFun192`, :cite:t:`Ehrhardt2018` (3.2.17.9).



    An example with real input for `z` and with purely imaginary input for `\tau`, producing real output:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; z = '100'; tau = '0.0 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_p_inv(z, tau); mx = mpm.weierstrass_p_inv(z, tau)
        >>> gx = gmp.weierstrass_p_inv(z, tau)
        >>> fx = fpm.weierstrass_p_inv(z, tau); ax = apm.weierstrass_p_inv(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  1.000602721184701332310197413726890982085E-1
        mpm:  1.000602721184701332310197413726890982085e-1
        gmp:  1.000602721184701332310197413726890982085E-01
        fpm:  1.00060272118470E-01
        apm:  1.000602721184701332310197413726890982085e-1 (7.17e-40%)




    An example with complex input for `z` and `\tau`:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 20; z = '0.1 + 0.5j'; tau = '0.7 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_p_inv(z, tau); mx = mpm.weierstrass_p_inv(z, tau)
        >>> gx = gmp.weierstrass_p_inv(z, tau)
        >>> fx = fpm.weierstrass_p_inv(z, tau); ax = apm.weierstrass_p_inv(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax], aligned=True)
        dec: 4.2630465458606458544E-1              - 3.0967048559182771279E-1j
        mpm: 4.2630465458606458544e-1              - 3.0967048559182771279e-1j
        gmp: 4.2630465458606458544E-01             - 3.0967048559182771279E-01j
        fpm: 4.26304654586065E-01                  - 3.09670485591828E-01j
        apm: 4.2630465458606458544e-1 (1.265e-16%) - 3.0967048559182771279e-1 (-1.063e-16%)j





    
|newpage|



.. _rst_mpm_weierstrass_zeta: 

Weierstrass Zeta function `\zeta_g(z, g_2, g_3)`
-------------------------------------------------------------------------------


.. method:: ctxflint.weierstrass_zeta_g(z, g2, g3)


    Computes the Weierstrass zeta function `\zeta_g(z; g_2, g_3)`. 

    We have `\zeta_g(tz; t^{-4} g_2, t^{-6} g_3) = t^{-1} \zeta_g(z; g_2, g_3)` and `\zeta_g(i z; g_2, g_3) = -i\zeta_g(z; g_2, -g_3)`. 


    The function is related to `\wp(z; g_2, g_3)` by `\displaystyle \frac{d \zeta(z; g_2, g_3)}{\mathrm{d}z} = -\wp(z; g_2, g_3)` and `\displaystyle \zeta(z; g_2, g_3) - z^{-1} = \int_0^z \left(\wp(z; g_2, g_3) - z^{-2} \right)`.


    See also MathWorld :cite:p:`WolframFun194`, Flint :cite:p:`FlintFun190`.

    

|13a_TestWeierstrassZeta_re| `\quad` |13b_TestWeierstrassZeta_im| `\quad` |13c_TestWeierstrassZeta_abs|

.. |13a_TestWeierstrassZeta_re| image:: ../_static/ExplicitSurfaces/CplxElliptic/13a_TestWeierstrassZeta_re.3D.xml.jpg
   :width: 30 %

.. |13b_TestWeierstrassZeta_im| image:: ../_static/ExplicitSurfaces/CplxElliptic/13b_TestWeierstrassZeta_im.3D.xml.jpg
   :width: 30 %

.. |13c_TestWeierstrassZeta_abs| image:: ../_static/ExplicitSurfaces/CplxElliptic/13c_TestWeierstrassZeta_abs.3D.xml.jpg
   :width: 30 %



**Left figure**: real part of the Weierstrass Zeta function `\zeta_g(z, g_2, g_3)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Middle figure**: imaginary part of the Weierstrass Zeta function `\zeta_g(z, g_2, g_3)`. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Right figure**:  absolute value of the Weierstrass Zeta function `\zeta_g(z, g_2, g_3)`, with color-coded phase. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.







    For `\Delta = g_2^3 - 27 g_3^2 \ne 0`, we first compute the elliptic half periods `\omega_1, \omega_2` from the lattice  invariants `g_2, g_3` (see :ref:`EllipticHalfPeriodsG <rst_mpm_elliptic_halfperiods_from_invariants>`), and set `\omega = \omega_1, \tau = \omega_2/\omega_1`. Then `\zeta_g(z; g_2, g_3) = \zeta(z| \omega, \tau)` (see :ref:`WeierstrassZeta <rst_mpm_weierstrass_zeta_by_tau>`).


    For `\Delta = 0, g_2 > 0, g_3 \ne 0`, the function `\zeta_g(\cdot)` can be expressed in closed form in terms of elementary functions:


    For `\Delta = 0, g_2 > 0, g_3 < 0: \quad  \displaystyle \zeta_g(z; g_2, g_3) = -cz + \sqrt{3c}  \coth( \sqrt{3c}z ),  \quad  \text{where } c = \sqrt{g_2/12}`.


    For `\Delta = 0, g_2 > 0, g_3 > 0: \quad  \displaystyle \zeta_g(z; g_2, g_3) = cz + \sqrt{3c}  \cot( \sqrt{3c}z ),  \quad  \text{where } c = \sqrt{g_2/12}`.


    If both `g_2` and `g_3` are zero, the function `\zeta_g(\cdot)` becomes `\zeta_g(z; 0, 0) = z^{-1}`.







    An example with real input for `z` and with purely imaginary input for `\tau`, producing real output:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; z = '0.1'; tau = '0.0 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_zeta(z, tau); mx = mpm.weierstrass_zeta(z, tau)
        >>> gx = gmp.weierstrass_zeta(z, tau)
        >>> fx = fpm.weierstrass_zeta(z, tau); ax = apm.weierstrass_zeta(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  9.995978010353406976694591385190992698486E+0
        mpm:  9.995978010353406976694591385190992698486e+0
        gmp:  9.995978010353406976694591385190992698486E+00
        fpm:  9.99597801035341E+00
        apm:  9.995978010353406976694591385190992698486e+0 (1.654e-38%)






|newpage|


.. _rst_mpm_weierstrass_sigma: 

Weierstrass Sigma function `\sigma_g(z, g_2, g_3`)
-------------------------------------------------------------------------------


.. method:: ctxflint.weierstrass_sigma_g(z, g2, g3)


    Computes the Weierstrass sigma function, `\sigma_g(z; g_2, g_3)`. We have `\sigma_g(tz; t^{-4} g_2, t^{-6} g_3) = t \sigma_g(z; g_2, g_3)`.  |newline|

    The function is related to `\zeta(z; g_2, g_3)` by `\displaystyle \frac{d}{\mathrm{d}z} \log \sigma(z; g_2, g_3) = \zeta(z; g_2, g_3)`.


    See also MathWorld :cite:p:`WolframFun193`, Flint :cite:p:`FlintFun190`.

    See also: https://dlmf.nist.gov/23.2


    


|14a_TestWeierstrassSigma_re| `\quad` |14b_TestWeierstrassSigma_im| `\quad` |14c_TestWeierstrassSigma_abs|

.. |14a_TestWeierstrassSigma_re| image:: ../_static/ExplicitSurfaces/CplxElliptic/14a_TestWeierstrassSigma_re.3D.xml.jpg
   :width: 30 %

.. |14b_TestWeierstrassSigma_im| image:: ../_static/ExplicitSurfaces/CplxElliptic/14b_TestWeierstrassSigma_im.3D.xml.jpg
   :width: 30 %

.. |14c_TestWeierstrassSigma_abs| image:: ../_static/ExplicitSurfaces/CplxElliptic/14c_TestWeierstrassSigma_abs.3D.xml.jpg
   :width: 30 %



**Left figure**: real part of the Weierstrass Sigma function `\sigma_g(z, g_2, g_3`). Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Middle figure**: imaginary part of the Weierstrass Sigma function `\sigma_g(z, g_2, g_3`). Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.


**Right figure**:  absolute value of the Weierstrass Sigma function `\sigma_g(z, g_2, g_3`), with color-coded phase. Camera angles are `\theta=135^\circ` and `\phi = -12^\circ`, camera radius is -2.





    For `\Delta = g_2^3 - 27 g_3^2 \ne 0`, we first compute the elliptic half periods `\omega_1, \omega_2` from the lattice  invariants `g_2, g_3` (see :ref:`EllipticHalfPeriodsG <rst_mpm_elliptic_halfperiods_from_invariants>`), and set `\omega = \omega_1, \tau = \omega_2/\omega_1`. Then `\sigma_g(z; g_2, g_3) = \sigma(z| \omega, \tau)` (see :ref:`WeierstrassSigma <rst_mpm_weierstrass_sigma_by_tau>`).


    For `\Delta = 0, g_2 > 0, g_3 \ne 0`, the function `\sigma_g(\cdot)` can be expressed in closed form in terms of elementary functions:


    For `\Delta = 0, g_2 > 0, g_3 < 0: \quad  \displaystyle \sigma_g(z; g_2, g_3) = \frac{\sinh( \sqrt{3c}z )}{\sqrt{3c} \cdot e^{cx^2 /2}},  \quad  \text{where } c = \sqrt{g_2/12}`.


    For `\Delta = 0, g_2 > 0, g_3 > 0: \quad  \displaystyle \sigma_g(z; g_2, g_3) = \frac{\sin( \sqrt{3c}z )}{\sqrt{3c} \cdot e^{cx^2 /2}},  \quad  \text{where } c = \sqrt{g_2/12}`.


    If both `g_2` and `g_3` are zero, the function `\sigma_g(\cdot)` becomes `\sigma_g(z; 0, 0) = z`.







    An example with real input for `z` and with purely imaginary input for `\tau`, producing real output:

    .. code-block:: pycon

        >>> from xlcalcnet import dec, mpm, gmp, fpm, apm
        >>> mpm.dps = 40; z = '0.1'; tau = '0.0 + 0.9j'
        >>> \mathrm{d}x = dec.weierstrass_sigma(z, tau); mx = mpm.weierstrass_sigma(z, tau)
        >>> gx = gmp.weierstrass_sigma(z, tau)
        >>> fx = fpm.weierstrass_sigma(z, tau); ax = apm.weierstrass_sigma(z, tau)
        >>> mpm.show([\mathrm{d}x, mx, gx, fx, ax])
        dec:  9.998992989830383649264433694319757846077E-2
        mpm:  9.998992989830383649264433694319757846077e-2
        gmp:  9.998992989830383649264433694319757846077E-02
        fpm:  9.99899298983038E-02
        apm:  9.998992989830383649264433694319757846077e-2 (5.023e-39%)