.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/01_ramanujan_master_theorem.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_01_ramanujan_master_theorem.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_01_ramanujan_master_theorem.py:


Ramanujan's Master Theorem
==========================

If :math:`f(x) = \sum_n \varphi(n) (-x)^n / n!`, then

.. math:: \int_0^\infty x^{s-1} f(x)\,dx = \Gamma(s)\,\varphi(-s).

This example applies the theorem to :math:`(1+x)^{-3}`, checks it against the
Beta function and a numerical Mellin transform, and evaluates a dimensionally
regulated radial integral, where the theorem *is* the analytic continuation in D.

.. GENERATED FROM PYTHON SOURCE LINES 13-29




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

 .. code-block:: none

    Int x^(s-1) (1+x)^-3 dx   RMT     : 0.416520275452346835657738842818
                              Beta    : 0.416520275452346835657738842818
                              numeric : 0.416520275452346835657738842818
    Int dq2 q2^(D/2-1)/(q2+1),  D=1: 3.14159265358979323846264338328
    Int dq2 q2^(D/2-1)/(q2+1),  D=3: -3.14159265358979323846264338328
    Int dq2 q2^(D/2-1)/(q2+1),  D=3.9: -20.0824840790797440454036816166






|

.. code-block:: Python

    import mpmath as mp

    from ramanujan_qft import massive_propagator_rmt, mellin_numeric, ramanujan_master_theorem

    mp.mp.dps = 30

    # 1/(1+x)^a = Sum (a)_n (-x)^n / n!   ->  phi(n) = Gamma(a+n)/Gamma(a)
    a, s = mp.mpf(3), mp.mpf("1.25")
    rmt = ramanujan_master_theorem(lambda n: mp.gamma(a + n) / mp.gamma(a), s)
    print("Int x^(s-1) (1+x)^-3 dx   RMT     :", rmt)
    print("                          Beta    :", mp.beta(s, a - s))
    print("                          numeric :", mellin_numeric(lambda x: (1 + x)**-a, s))

    # A dimensionally regulated radial integral: the RMT result *is* the analytic continuation in D.
    for D in (1, 3, mp.mpf("3.9")):
        print(f"Int dq2 q2^(D/2-1)/(q2+1),  D={mp.nstr(D, 3)}:", massive_propagator_rmt(1, D))


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

   **Total running time of the script:** (0 minutes 0.019 seconds)


.. _sphx_glr_download_auto_examples_01_ramanujan_master_theorem.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: 01_ramanujan_master_theorem.ipynb <01_ramanujan_master_theorem.ipynb>`

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

      :download:`Download Python source code: 01_ramanujan_master_theorem.py <01_ramanujan_master_theorem.py>`

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

      :download:`Download zipped: 01_ramanujan_master_theorem.zip <01_ramanujan_master_theorem.zip>`


.. only:: html

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

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