feat: add numerical laplace transform

Author: Tushar-R-Tyagi

maths/laplace_transformation.py

@@ -0,0 +1,63 @@
+"""
+The Laplace Transform is defined as: L{f(t)} = integral from 0 to infinity of e^(-st) * f(t) dt.
+
+Wiki: https://en.wikipedia.org/wiki/Laplace_transform
+
+"""
+
+import numpy as np
+
+
+def laplace_transform(
+    function_values: np.ndarray, s_value: float, delta_t: float
+) -> float:
+    """
+    Calculate the numerical Laplace Transform of a function given its values over time.
+
+    Args:
+        function_values: A numpy array of the function values f(t).
+        s_value: The complex frequency parameter 's' (modeled here as a float).
+        delta_t: The time step between samples.
+
+    Returns:
+        The approximate 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
+    """
+    if s_value < 0:
+        raise ValueError("s_value must be non-negative for convergence.")
+
+    # Time vector corresponding to the function values
+    time_vector = np.arange(len(function_values)) * delta_t
+    
+    # The integrand: f(t) * e^(-s*t)
+    integrand = function_values * np.exp(-s_value * time_vector)
+    
+    # Numerical integration using the trapezoidal rule
+    result = np.trapezoid(integrand, dx=delta_t)
+    
+    return float(result)
+
+
+if __name__ == "__main__":
+    import doctest
+
+    
+    doctest.testmod()
+