maths.laplace_transformation
============================

.. py:module:: maths.laplace_transformation

.. autoapi-nested-parse::

   Laplace Transform — Numerical Implementation.

   Computes the numerical Laplace Transform using the trapezoidal
   integration rule. Supports real-valued, non-negative Laplace
   parameters only.

   Reference: https://en.wikipedia.org/wiki/Laplace_transform



Functions
---------

.. autoapisummary::

   maths.laplace_transformation.laplace_transform


Module Contents
---------------

.. py:function:: laplace_transform(function_values: numpy.ndarray, s_value: float, delta_t: float) -> float

   Calculate the numerical Laplace Transform of a function given its values over time.

   This implementation supports only real-valued, non-negative Laplace
   parameters ``s``.

   Args:
       function_values: A numpy array of the function values f(t).
       s_value: The real-valued Laplace parameter ``s``. Must be non-negative.
       delta_t: The time step between samples.

   Returns:
       The approximate real-valued value of the Laplace transform at s_value.

   Example: For f(t) = 1, the Laplace transform L{1} = 1/s.
   If s = 2, L{1} should be 0.5.

   >>> t = np.linspace(0, 50, 10000)
   >>> f_t = np.ones_like(t) # f(t) = 1
   >>> res = laplace_transform(f_t, s_value=2.0, delta_t=50/10000)
   >>> abs(res - 0.5) < 1e-3
   True

   Example: For f(t) = e^(-t), the Laplace transform L{e^-t} = 1/(s+1).
   If s = 1, L{e^-t} should be 0.5.

   >>> t = np.linspace(0, 50, 10000)
   >>> f_t = np.exp(-t)
   >>> res = laplace_transform(f_t, s_value=1.0, delta_t=50/10000)
   >>> abs(res - 0.5) < 1e-3
   True