:orphan:

Examples
========

Each example is a self-contained script; the output shown below each one was
produced when this documentation was built, with the released version of the
package. Download any script with the link at the bottom of its page.


.. 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="The imaginary part of the massive sunset from its phase-space representation, and the twice-subtracted dispersion relation S(s) - S(0) - s S&#x27;(0). Both are precision-checked numerical integrations, validated against Feynman-parameter integrals and a closed form. Each subtracted value takes 10-20 s.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_04_sunset_dispersion_thumb.png
    :alt:

  :doc:`/auto_examples/04_sunset_dispersion`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Two-loop sunset: imaginary part and dispersion relation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A separate Taylor method fills the equal-mass gap in the RMT charts.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_07_sunset_taylor_thumb.png
    :alt:

  :doc:`/auto_examples/07_sunset_taylor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Equal-mass vacuum and Euclidean continuation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="If f(x) = \sum_n \varphi(n) (-x)^n / n!, then">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_01_ramanujan_master_theorem_thumb.png
    :alt:

  :doc:`/auto_examples/01_ramanujan_master_theorem`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Ramanujan's Master Theorem</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Scalar B_0 and C_0 in LoopTools conventions, to double precision and to 50 digits, and the Method-of-Brackets series for B_0 compared with its closed form.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_03_one_loop_integrals_thumb.png
    :alt:

  :doc:`/auto_examples/03_one_loop_integrals`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">One-loop integrals at any precision</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Exact bracket algebra, analytic pole resolution, and convergent summation. This example uses no numerical integration backend.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_06_sunset_rmt_thumb.png
    :alt:

  :doc:`/auto_examples/06_sunset_rmt`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">A graph-generated multivariate RMT series</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Graph construction, RMT scale integration, and numerical Hepp sectors. Requires pip install &#x27;.[multiloop]&#x27;. Errors are statistical estimates.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_05_multiloop_graphs_thumb.png
    :alt:

  :doc:`/auto_examples/05_multiloop_graphs`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Massive Euclidean multiloop graphs</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="\int_0^\infty e^{-x} J_0(w x)\,dx = 1/\sqrt{1+w^2}. For large w the integrand oscillates many times per unit length, while the residue series converges like (1/w)^n. The example also shows a logarithmic case (coinciding poles), a three-factor product with no closed form, and the engine refusing a case at its convergence boundary instead of guessing.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_02_mellin_products_thumb.png
    :alt:

  :doc:`/auto_examples/02_mellin_products`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Product integrals by Mellin residue summation</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A scalar one-loop integral, a coupled RMT series, and a hybrid three-loop graph. Install the multiloop extra. Mass inputs below are squared masses.">

.. only:: html

  .. image:: /auto_examples/images/thumb/sphx_glr_08_first_calculations_thumb.png
    :alt:

  :doc:`/auto_examples/08_first_calculations`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Three calculations with independent references</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /auto_examples/04_sunset_dispersion
   /auto_examples/07_sunset_taylor
   /auto_examples/01_ramanujan_master_theorem
   /auto_examples/03_one_loop_integrals
   /auto_examples/06_sunset_rmt
   /auto_examples/05_multiloop_graphs
   /auto_examples/02_mellin_products
   /auto_examples/08_first_calculations



.. only:: html

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

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