.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/special_functions/gamma_beta/plot_02_stirling_formula.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_special_functions_gamma_beta_plot_02_stirling_formula.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_special_functions_gamma_beta_plot_02_stirling_formula.py:


Stirling's formula and its asymptotic series
===============================================

Shows that sqrt(2 pi n) (n/e)^n approximates n! with relative error
about 1/(12n), and that Stirling's full series for log Gamma(x) is
asymptotic: at small x, adding terms first helps and then hurts.

.. GENERATED FROM PYTHON SOURCE LINES 11-18

.. code-block:: Python

    import math

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.special_functions import log_gamma_function, stirling_factorial, stirling_log_gamma








.. GENERATED FROM PYTHON SOURCE LINES 19-21

The leading term: n! ~ sqrt(2 pi n) (n/e)^n
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-26

.. code-block:: Python


    for n in (1, 5, 10, 50, 100):
        ratio = math.factorial(n) / stirling_factorial(float(n))
        print(f"n = {n:3d}: n!/Stirling = {ratio:.6f}, 1 + 1/(12n) = {1 + 1 / (12 * n):.6f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    n =   1: n!/Stirling = 1.084438, 1 + 1/(12n) = 1.083333
    n =   5: n!/Stirling = 1.016784, 1 + 1/(12n) = 1.016667
    n =  10: n!/Stirling = 1.008365, 1 + 1/(12n) = 1.008333
    n =  50: n!/Stirling = 1.001668, 1 + 1/(12n) = 1.001667
    n = 100: n!/Stirling = 1.000834, 1 + 1/(12n) = 1.000833




.. GENERATED FROM PYTHON SOURCE LINES 27-29

Error of the truncated series versus number of terms
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 29-40

.. code-block:: Python


    terms = np.arange(0, 16)
    fig, ax = plt.subplots()
    for x in (1.0, 2.0, 5.0):
        errors = [abs(stirling_log_gamma(x, terms=int(k)) - log_gamma_function(x)) for k in terms]
        ax.semilogy(terms, errors, "o-", label=f"x = {x:g}")
        print(f"x = {x:g}: best with {int(np.argmin(errors))} terms, error {min(errors):.2e}")
    ax.set_xlabel("number of Bernoulli correction terms")
    ax.set_ylabel(r"$|\,\mathrm{series} - \ln\Gamma(x)\,|$")
    ax.set_title("Stirling's series is asymptotic, not convergent")
    ax.legend()



.. image-sg:: /api/gallery/special_functions/gamma_beta/images/sphx_glr_plot_02_stirling_formula_001.png
   :alt: Stirling's series is asymptotic, not convergent
   :srcset: /api/gallery/special_functions/gamma_beta/images/sphx_glr_plot_02_stirling_formula_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    x = 1: best with 3 terms, error 2.88e-04
    x = 2: best with 7 terms, error 3.90e-07
    x = 5: best with 15 terms, error 8.88e-16

    <matplotlib.legend.Legend object at 0x11c4ff4d0>




.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.043 seconds)


.. _sphx_glr_download_api_gallery_special_functions_gamma_beta_plot_02_stirling_formula.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_stirling_formula.ipynb <plot_02_stirling_formula.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_stirling_formula.py <plot_02_stirling_formula.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_stirling_formula.zip <plot_02_stirling_formula.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_