Distributional Derivative of ( f(x) = x^\lambda ) for ( x > 0 ): Theory & SageMath Implementation

Distributional Derivative of \( f(x)=x^λ \) for 𝑥>0: Theoretical Insights and SageMath Implementation

1. Introduction

In this post, we dive deeper into the world of generalized (distributional) derivatives and examine how singularities affect the process of differentiation.
We focus on the function: \[ f(x) = x^\lambda \], for 𝑥>0 , with \( \quad -1 < \lambda < 0 \)
This function is locally summable, but its classical derivative: \[ f'(x) = \lambda x^{\lambda - 1} \] is not integrable near 𝑥=0, and hence not a true function in the sense of distributions. To handle this, we need a regularization approach to define its derivative in a meaningful distributional sense.
Can differentiation be redefined at singular points? What happens when classical differentiation fails? Let's explore these ideas, both theoretically and computationally using SageMath.

2. Theoretical Foundation

2.1 Functional Derivative Definition

In the theory of distributions, we define the derivative of a function \( 𝑓(𝑥) \)via its action on a test function \( \varphi(x) \) \[ (f', \varphi) = -\int_{0}^{\infty} f(x) \varphi'(x) ,dx \] For \( f(x)= x^\lambda \), this become: \[ (f', \varphi) = -\int_{0}\lambda \varphi'(x) ,dx \] This integral diverges at 𝑥=0 if \( 𝜆 < 0 \) , so we apply a regularization technique by subtracting the singularity at the origin: \[ \varphi(x) \rightarrow \varphi(x) - \varphi(0) \] Now the integral becomes: \[ (f', \varphi) = -\int_{0}\lambda (\varphi'(x) - \varphi'(0)) ,dx\] This regularized form defines a distribution and justifies the generalized derivative as: \[ (x^\lambda)' = \lambda x^{\lambda - 1} \] ,valid only when paired with test functions vanishing at x=0

3. SageMath Implementation

We now use SageMath to verify and visualize the behavior of this distributional derivative.

3.1 Define the Functionn

      
from sympy import Heaviside
from sage.all import var, integrate, diff

# Declare variables with valid names
x, lam = var('x lam')  # Changed 'lambda' to 'lam'

# Define f(x) = x^lam * Heaviside(x)
f_x = x**lam * Heaviside(x)

print("Function f(x):", f_x)
	
    

3.2 Compute Generalized Derivative

      
# Symbolic derivative using Sage
f_prime = diff(f_x, x)

print("Symbolic derivative f'(x):", f_prime)
	
    

This result reveals the Dirac delta behavior at 𝑥=0, capturing the singular nature of the derivative. The term 𝛿(𝑥) ensures the correctness of the derivative in the distributional framework.

3.3 Functional Regularization (Integration by Parts)

We now apply integration by parts to compute the generalized derivative manually:

      
phi = var('phi')  # Test function placeholder

# Functional form of generalized derivative
regularized_f_prime = -integrate(x**lam * diff(phi, x), x, 0, +oo)

print("Regularized functional:", regularized_f_prime)
	
    

This expression represents the functional action of the derivative on a test function 𝜑(𝑥), and corresponds to the distributional interpretation.

4. Visualization

To visualize the function \( f(x) = x^\lambda \cdot \theta(x) \) for different values of 𝜆, try:

      
from sympy import Heaviside
from sage.all import plot, var

x = var('x')

# Define the function
f_x = x**0.5 * Heaviside(x)

# Restrict the domain to x ≥ 0
plot(f_x, (x, 0, 5), title="f(x) = x^0.5 * Heaviside(x)")
	
    

📊 Try different values:

• λ=−0.5 → singular at origin
• λ=1, λ=2 → smooth behavior

This helps understand how the function behaves near the singularity at 𝑥=0.

5. Conclusion

• The function \( f(x) = x^\lambda \) is not classically differentiable at 𝑥=0 for 𝜆<0, but its distributional derivative is well-defined.
• Regularization is essential to make sense of otherwise divergent expressions.
• SageMath is a powerful tool for computing and visualizing such expressions, offering both symbolic and functional insights.

Next upcoming

"Beyond \(x^\lambda \): Exploring Regularization Methods for Singular Integrals in Generalized Functions"

• Logarithmic singularities \( f(x)=ln∣x∣ \)
• Principal value integrals
• Distributions with non-integrable behavior

💡 Try It Yourself! Now You can copy and paste directly into here Run SageMath Code Here