.. _sphx_glr_api_gallery_special_functions_laplace_transform:

Laplace transform
-----------------

The Laplace transform by quadrature, and its numerical inversion.


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    <div class="sphx-glr-thumbcontainer" tooltip="Laplace&#x27;s integral F(s) = int_0^inf f(t) e^{-st} dt turns differentiation into multiplication by s. This example computes the transform numerically for a few classical pairs, and uses the derivative rule to solve a damped oscillator algebraically.">

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  .. image:: /api/gallery/special_functions/laplace_transform/images/thumb/sphx_glr_plot_01_laplace_transform_pairs_thumb.png
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  :doc:`/api/gallery/special_functions/laplace_transform/plot_01_laplace_transform_pairs`

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      <div class="sphx-glr-thumbnail-title">Laplace's transform: from functions of t to functions of s</div>
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    <div class="sphx-glr-thumbcontainer" tooltip="Stehfest&#x27;s 1970 algorithm inverts F(s) from real samples only, with alternating weights that cost digits to cancellation. Talbot&#x27;s 1979 contour winds the Bromwich integral around the negative real axis, where e^{st} decays, and reaches near machine precision, even for oscillating f(t) that Stehfest cannot resolve.">

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  .. image:: /api/gallery/special_functions/laplace_transform/images/thumb/sphx_glr_plot_02_talbot_stehfest_inversion_thumb.png
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  :doc:`/api/gallery/special_functions/laplace_transform/plot_02_talbot_stehfest_inversion`

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      <div class="sphx-glr-thumbnail-title">Numerical Laplace inversion: Stehfest and Talbot</div>
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.. toctree::
   :hidden:

   /api/gallery/special_functions/laplace_transform/plot_01_laplace_transform_pairs
   /api/gallery/special_functions/laplace_transform/plot_02_talbot_stehfest_inversion