.. _sphx_glr_api_gallery_special_functions_orthogonal_polynomials:

Orthogonal polynomials
------------------------

Legendre and Chebyshev polynomials on [-1, 1], and Hermite and Laguerre
polynomials on infinite domains, with numerically verified orthogonality.


.. raw:: html

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


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Plots the first Legendre polynomials P_n (weight 1) beside the first Chebyshev polynomials T_n (weight 1/sqrt(1 - x^2)), checks each family&#x27;s orthogonality, and confirms Chebyshev&#x27;s closed form T_n(cos t) = cos(nt), whose equal ripples between -1 and 1 are its minimal-oscillation property.">

.. only:: html

  .. image:: /api/gallery/special_functions/orthogonal_polynomials/images/thumb/sphx_glr_plot_01_legendre_chebyshev_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/orthogonal_polynomials/plot_01_legendre_chebyshev`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Legendre and Chebyshev polynomials on [-1, 1]</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Plots the first Hermite polynomials H_n (weight e^{-x^2} on the whole line) and Laguerre polynomials L_n (weight e^{-x} on the half-line), checks their orthogonality over infinite intervals, and draws the quantum harmonic oscillator states H_n(x) e^{-x^2/2} they describe.">

.. only:: html

  .. image:: /api/gallery/special_functions/orthogonal_polynomials/images/thumb/sphx_glr_plot_02_hermite_laguerre_thumb.png
    :alt:

  :doc:`/api/gallery/special_functions/orthogonal_polynomials/plot_02_hermite_laguerre`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Hermite and Laguerre polynomials on infinite domains</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/special_functions/orthogonal_polynomials/plot_01_legendre_chebyshev
   /api/gallery/special_functions/orthogonal_polynomials/plot_02_hermite_laguerre