Complex Generalized Functions: Theory, Computation & Visualization with SageMath

Understanding Complex Generalized Functions: Extending Classical Analysis

Introduction

Classical analysis often stumbles when faced with singularities or discontinuities — think of point charges in physics or abrupt signal jumps in engineering. This is where generalized functions or distributions step in, extending classical functions to capture such irregular behaviors rigorously.

While distribution theory is well-studied over real domains, many modern applications — from quantum mechanics to signal processing — demand an extension to the complex domain. Complex generalized functions allow us to handle wavefunctions, complex-valued signals, and operators involving complex variables with mathematical precision.

This post aims to:

• Introduce the theory behind complex generalized functions,
• Illustrate their fundamental properties,
• And demonstrate how to compute and visualize them using SageMath, an open-source mathematics software system.

Throughout the post, you’ll find interactive SageMath cells — we encourage you to modify the functions and explore the beautiful interplay between theory and computation!

1.Theoretical Foundations

1.1 Complex Test Function Space

In distribution theory, the starting point is the space of test functions — smooth functions with compact support. For the complex case, define \[ K := \{ \varphi : \mathbb{R} \to \mathbb{C} \mid \varphi \text{ is infinitely differentiable with compact support} \} \]

These are just like the usual test functions, except now their values lie in the complex numbers 𝐶. This extension preserves the smoothness and compactness properties crucial for distribution theory.

1.2 Defining Complex Generalized Functions

A complex generalized function 𝑓∈𝐾′ is a continuous linear functional acting on 𝐾: \[ f: K \to \mathbb{C}, \quad \varphi \mapsto (f, \varphi). \] For example, if 𝑓(𝑥) is a complex locally integrable function, we define its associated functional by \[ (f, \varphi) = \int_{-\infty}^{\infty} f(x) \varphi(x) \,dx. \] Note: the complex structure appears in the values of 𝑓 and 𝜑, but the domain remains \( \mathbb{R} \).

1.3 Basic Operations

These operations extend naturally from the real-valued case:

• Addition and scalar multiplication ( \( \alpha \in \mathbb{C} \)):\[ (f+g, \varphi) = (f, \varphi) + (g, \varphi), \quad (\alpha f, \varphi) = \alpha (f, \varphi). \]
• Multiplication by a smooth complex function ( \( \alpha(x) \in \mathbb{C}^{\infty}(\mathbb{R},\mathbb{C} \)):\[ \quad (af, \varphi) := (f, \overline{a} \varphi), \] where \( \overline{a} \) is the complex conjugate of 𝑎. The conjugation here ensures compatibility with the inner product structure in complex Hilbert spaces.
• Complex conjugation: \[ \quad (\overline{f}, \varphi) := (\overline{f, \varphi}) \] For classical functions, this corresponds to the pointwise complex conjugate.

2. Visualizing Complex Test Functions with SageMath

Understanding complex functions is much easier when we visualize their real and imaginary parts.

      
# Define symbolic variable
var("x")

# Define a complex test function
phi(x) = exp(-x^2) + I*sin(x)

# Plot real and imaginary parts
plot_real = plot(real(phi(x)), (x, -5, 5), color='blue', legend_label='Real part')
plot_imag = plot(imag(phi(x)), (x, -5, 5), color='red', legend_label='Imaginary part')

# Show combined plot
show(plot_real + plot_imag, title='Real and Imaginary Components of φ(x)')
	
    

Try it live in Run SageMath Code Here Feel free to modify the function 𝜑(𝑥) in the code and observe how the plot changes!

Caption: The plot illustrates the real (blue) and imaginary (red) components of the complex test function \( \varphi(x) = e^{-x^2} + i \sin(x) \)

3. Computational Exploration with SageMath

3.1 Defining the Generalized Function Pairing

We now define a SageMath function to compute the pairing (𝑓,𝜑) for complex functions 𝑓 and 𝜑.

      
var("x")

def generalized_function(f, p):
    return integrate(f(x) * conjugate(p(x)), x, -Infinity, Infinity)
	
    

3.2 Multiplication by a Complex Function (Computational Verification)

Let’s verify the property \[ \quad (af, \varphi) := (f, \overline{a} \varphi), \]

      
# Define symbolic variable
var("x")

# Proper function definition for symbolic expressions
phi = function("phi")(x)
a = function("a")(x)
f = function("f")(x)

# Define test function values
phi = exp(-x^2) + I*sin(x)
a = cos(x) + I*sin(x)
f = 1 + I

# Define generalized function pairing
def generalized_function(f_expr, p_expr):
    return integrate(f_expr * conjugate(p_expr), (x, -Infinity, Infinity))

# Compute left-hand and right-hand sides
lhs = generalized_function(a * f, phi)
rhs = generalized_function(f, conjugate(a) * phi)

print("Left-hand side: ", lhs)
print("Right-hand side: ", rhs)

# Check equality
show(lhs == rhs)
	
    

Try this: Run SageMath Code Here Just copy the code and run it here to see the result.

Your integral is diverging, meaning it does not converge properly over the given range (-∞ to ∞).

Solutions & Fixe

• Use Finite Integration Limits
• Plot the Function to Check Behavior
• Try Numerical Integration Instead
• Apply Regularization

Example

      
# Define symbolic variable
var("x")

# Proper function definition for symbolic expressions
phi = function("phi")(x)
a = function("a")(x)
f = function("f")(x)

# Define test function values
phi = exp(-x^2) + I*sin(x)
a = cos(x) + I*sin(x)
f = 1 + I

# Define generalized function pairing
def generalized_function(f_expr, p_expr):
    return integrate(f_expr * conjugate(p_expr), (x, -Infinity, Infinity))
    
real_part = real(phi * conjugate(a) * f)
imag_part = imag(phi * conjugate(a) * f)

num_real = numerical_integral(real_part, -10, 10)
num_imag = numerical_integral(imag_part, -10, 10)

print("Numerical Real Integral:", num_real)
print("Numerical Imaginary Integral:", num_imag)

symbolic_result = integrate(real_part + I * imag_part, (x, -10, 10))
print(symbolic_result)

    

Try this: Run SageMath Code Here Just copy the code and run it here to see the result.

🔍 Interpretation of Results
Numerical Real Integral: ≈ 10.92 (with negligible error ~10⁻¹³)
Numerical Imaginary Integral: ≈ 10.92 (also with negligible error ~10⁻¹³)
Since both real and imaginary integrals are nearly identical, this suggests symmetry or proportionality in the function behavior.

Step 1: Visualizing the Function Behavior

To better understand why your symbolic and numerical results are similar, let’s plot the real and imaginary parts:

      
plot_total = (plot(real(phi * conjugate(a) * f), (x, -10, 10), color='blue', legend_label='Real Part') +
              plot(imag(phi * conjugate(a) * f), (x, -10, 10), color='red', legend_label='Imaginary Part'))
show(plot_total)

    

• - This helps confirm whether the real and imaginary components behave similarly.
• - If they follow similar trends, their integrals will also be nearly equal.

Step 2: Comparing Symbolic vs. Numerical Result

Let’s directly compare the numerical and symbolic results:

      
symbolic_eval = integrate(real(phi * conjugate(a) * f) + I * imag(phi * conjugate(a) * f), (x, -10, 10))
num_eval = num_real[0] + I * num_imag[0]

print("Symbolic Evaluation:", symbolic_eval)
print("Numerical Evaluation:", num_eval)
print("Difference:", symbolic_eval - num_eval)

    

• - If the difference is small, your symbolic and numerical integrations are consistent.
• - If the difference is large, it suggests symbolic integration may have additional terms or rounding errors affecting results

Step 3: Investigate Why Symbolic Results Contain Exponential & Error Functions

Your symbolic result includes terms like e^(1/4), sqrt(pi), and erf(...)—this is common when integrating Gaussian-like functions.

      
simplify(symbolic_result)

    

• If the symbolic result reduces to something closer to 10.92 + I*10.92, we confirm numerical and symbolic consistency.

3.3 Complex Conjugation (Computational Verification)

Verify that \[ \quad (\overline{f}, \varphi) := (\overline{f, \varphi}) \]

      
var("x")

def generalized_function(f_expr, p_expr):
    return integrate(f_expr(x=x) * conjugate(p_expr(x=x)), x, -Infinity, Infinity)

f(x) = (1 + I) * x * exp(-x^2)
phi(x) = exp(I * x) * exp(-x^2)

conj_f_integral = generalized_function(conjugate(f), phi)
f_conj_phi_integral_conj = conjugate(generalized_function(f, conjugate(phi)))

print("(conjugate(f), phi): ", conj_f_integral)
print("conjugate((f, conjugate(phi))): ", f_conj_phi_integral_conj)
show(conj_f_integral == f_conj_phi_integral_conj)

    

Try this: Run SageMath Code Here Just copy the code and run it here to see the result.

Results indicate that the two expressions are equal, which confirms the expected complex conjugation property for generalized function pairing.

4. Applications of Complex Generalized Functions

• Quantum Mechanics: Wavefunctions are complex-valued and often described by distributions — for example, the delta distribution models point potentials or eigenstates. Complex generalized functions rigorously handle such states and operators.
• Signal Processing: Complex exponentials appear in Fourier analysis. Generalized functions help analyze singular or impulsive signals, enabling precise descriptions of filtering and modulation in the complex frequency domain.
• Mathematical Physics: Complex-valued Green’s functions (distributional solutions) solve PDEs in electromagnetism and quantum field theory. Their distributional nature allows us to manage singularities and boundary conditions elegantly.

5. Extensions and Further Considerations

Most real distribution theory results extend to the complex case with slight adaptations. For example, notions of continuity, differentiation, and support still hold, but complex conjugation and multiplication require careful attention.

Future explorations might include:

• Complex Fourier transforms of distributions,
• Holomorphic distributions,
• Applications in several complex variables and complex analysis.

6. Conclusion

Extending generalized functions to the complex domain vastly broadens the toolkit of analysts, physicists, and engineers tackling singular or irregular complex phenomena. The marriage of rigorous theory and computational tools like SageMath makes these advanced concepts approachable and interactive.

We encourage you to tinker with the SageMath cells embedded throughout this post — experiment with your own functions and see theory come alive!

7. Recommended Readings

• L. Schwartz, Theory of Distributions, 1950.
• I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 1: Properties and Operations, 1964.