.. _sphx_glr_api_gallery_special_functions_elliptic:

Elliptic functions
------------------

Elliptic integrals, the arithmetic-geometric mean, and Jacobi elliptic functions.


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  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Iterates the AGM by hand to show its quadratic convergence, then checks Gauss&#x27;s 1799 identity K(m) = pi / (2 AGM(1, sqrt(1 - m))) and uses E(m) for the perimeter of an ellipse.">

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  .. image:: /api/gallery/special_functions/elliptic/images/thumb/sphx_glr_plot_01_agm_elliptic_integrals_thumb.png
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  :doc:`/api/gallery/special_functions/elliptic/plot_01_agm_elliptic_integrals`

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      <div class="sphx-glr-thumbnail-title">Gauss's arithmetic-geometric mean and the elliptic integral K</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Plots sn, cn, dn for m = 0.9, showing their real period 4K(m), and uses them to write the exact motion of a large-amplitude pendulum.">

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  .. image:: /api/gallery/special_functions/elliptic/images/thumb/sphx_glr_plot_02_jacobi_elliptic_functions_thumb.png
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  :doc:`/api/gallery/special_functions/elliptic/plot_02_jacobi_elliptic_functions`

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      <div class="sphx-glr-thumbnail-title">Jacobi elliptic functions and the pendulum</div>
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.. toctree::
   :hidden:

   /api/gallery/special_functions/elliptic/plot_01_agm_elliptic_integrals
   /api/gallery/special_functions/elliptic/plot_02_jacobi_elliptic_functions