Merge branch 'master' of https://github.com/Tushar-R-Tyagi/Python

Author: Tushar-R-Tyagi

maths/laplace_transformation.py

@@ -12,23 +12,32 @@ 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.
+    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'.
-        delta_t: The positive time step between samples.
+        s_value: The real-valued Laplace parameter ``s``. Must be non-negative.
+        delta_t: The time step between samples.
 
     Returns:
-        The approximate real-valued Laplace transform at s_value.
+        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, endpoint=False)
-    >>> f_t = np.ones_like(t)
+    >>> 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
 
-    >>> t = np.linspace(0, 50, 10000, endpoint=False)
+    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
@@ -45,8 +54,10 @@ def laplace_transform(
     # The integrand: f(t) * e^(-s*t)
     integrand = function_values * np.exp(-s_value * time_vector)
 
-    # Numerical integration using the trapezoid rule
-    return float(np.trapezoid(integrand, dx=delta_t))
+    # Numerical integration using the trapezoidal rule
+    result = np.trapezoid(integrand, dx=delta_t)
+
+    return float(result)
 
 
 if __name__ == "__main__":