refactor: add input validation and fix doctest precision

Author: Tushar-R-Tyagi

maths/laplace_transformation.py

@@ -1,7 +1,7 @@
 """
-The Laplace Transform is defined as: L{f(t)} = integral from 0 to infinity of e^(-st) * f(t) dt.
+This module provides a numerical implementation of the Laplace Transform.
 
-Wiki: https://en.wikipedia.org/wiki/Laplace_transform
+https://en.wikipedia.org/wiki/Laplace_transform
 
 """
 
@@ -12,52 +12,44 @@ 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.
+    Calculate the numerical Laplace Transform of a function given its values.
 
     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.
+        s_value: The real-valued Laplace parameter 's'.
+        delta_t: The positive time step between samples.
 
     Returns:
-        The approximate value of the Laplace transform at s_value.
+        The approximate real-valued 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
+    >>> t = np.linspace(0, 50, 10000, endpoint=False)
+    >>> f_t = np.ones_like(t)
     >>> 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)
+    >>> t = np.linspace(0, 50, 10000, endpoint=False)
     >>> 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.")
+    if delta_t <= 0:
+        raise ValueError("delta_t must be a positive value.")
+    if function_values.size == 0:
+        raise ValueError("function_values array cannot be empty.")
 
     # 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)
+
+    # Numerical integration using the trapezoid rule
+    return float(np.trapezoid(integrand, dx=delta_t))
 
 
 if __name__ == "__main__":
     import doctest
 
-    
     doctest.testmod()
-