Appendix. Weierstrass :math:`\wp` function and lattice sums
===========================================================

.. note::
   This section is under constant development. Its content will change and improve over time.

.. code:: ipython3

    import numpy as np

.. code:: ipython3

    from basicsums.weierstrass import Weierstrass
    from basicsums.eisenstein import E_numeric

.. code:: ipython3

    w1 = 1
    w2 = 1j

.. code:: ipython3

    wp = Weierstrass(w1/2, w2/2)

.. code:: ipython3

    wp.wp(0.1+0.1j) # Weierstrass P function




.. parsed-literal::

    (-4.440892098500626e-15-49.81116546459035j)



.. code:: ipython3

    wp.wpp(0.1+0.1j) # derivative of th Weierstrass P function




.. parsed-literal::

    (501.88358442957417+501.8835844295741j)



Eisenstein functions and Eisenstein-Rayleigh lattice sums
---------------------------------------------------------

.. code:: ipython3

    from basicsums.eisenstein import lattice_sums

.. code:: ipython3

    S = lattice_sums(w1, w2, 6)

.. code:: ipython3

    S[2]




.. parsed-literal::

    3.1415926535897927



.. code:: ipython3

    from basicsums.eisenstein import E

.. code:: ipython3

    E(3), E(4)




.. parsed-literal::

    (-P'(z)/2, P(z)**2 - 5*S(4))



.. code:: ipython3

    E(10)




.. parsed-literal::

    (8*P(z)**5 - 76*P(z)**3*S(4) + 5*P(z)**2*P'(z)**2 + 180*P(z)*S(4)**2 - 11*P'(z)**2*S(4))/28



.. code:: ipython3

    from basicsums.eisenstein import E_numeric

.. code:: ipython3

    E2 = E_numeric(2, w1, w2)
    E2(0.1 + 0.5j)




.. parsed-literal::

    (-2.8498259468658538-6.84841378126949e-16j)



.. code:: ipython3

    E3 = E_numeric(3, w1, w2)
    E3(0.1 + 0.5j)




.. parsed-literal::

    (-8.25402443117664+3.1086244689504383e-15j)



.. code:: ipython3

    E4 = E_numeric(4, w1, w2)
    E4(0.1 + 0.5j)




.. parsed-literal::

    (20.14103683511636+8.206342742542959e-15j)