1. What is the Product Log Function?

The Product Log function (also called Lambert W function) is defined as the inverse relation of the function f(W) = We^W. For any complex number z, it satisfies z = W(z)e^W(z).

2. How Does the Calculator Work?

The calculator solves the equation:

Implementation: The calculator uses Newton-Raphson iteration to numerically approximate the solution:

1. Start with initial approximation (different for x < 1 and x ≥ 1)
2. Iteratively refine using: \( W_{new} = W - \frac{We^W - x}{e^W(1 + W)} \)
3. Stop when difference between iterations is very small

3. Applications of the Lambert W Function

Applications include:

• Solving delay differential equations
• Analysis of algorithms and combinatorics
• Quantum mechanics (solutions to Schrödinger equation)
• Population growth models
• Electrical engineering (diode current-voltage relations)

4. Using the Calculator

Tips:

• Input must be ≥ -1/e (≈ -0.3679)
• For x ≥ 0, there is exactly one real solution
• For -1/e ≤ x < 0, there are two real solutions (calculator returns the principal branch)

5. Frequently Asked Questions (FAQ)

Q1: Why is the function called "Product Log"?
A: The name comes from its relationship to the logarithm and its product-like behavior in We^W.

Q2: What are the branches of the W function?
A: The function has multiple branches (like the complex logarithm). The calculator returns the principal branch (W₀).

Q3: What's the derivative of W(x)?
A: \( \frac{dW}{dx} = \frac{W(x)}{x(1 + W(x))} \) for x ≠ 0 and x ≠ -1/e

Q4: Can the calculator handle complex numbers?
A: This version only calculates real values. Complex W requires more sophisticated algorithms.

Q5: What's the Taylor series for W near x=0?
A: \( W(x) = \sum_{n=1}^\infty \frac{(-n)^{n-1}}{n!}x^n \) = x - x² + (3/2)x³ - (8/3)x⁴ + ...