Lambert W Real Branches

The Lambert W function is defined as the multi-valued inverse of the function w ↦ w e^w. Since the mapping is not injective on the real line, its real-valued inverse requires two branches.

Theory of Inverse Branches

The relation x = w e^w has a global minimum at w = -1, where x = -1/e. For x < -1/e, there are no real solutions for w. For -1/e ≤ x < 0, there are two real solutions for w: one in [-1, ∞) (the principal branch, W_0) and one in (-∞, -1] (the lower branch, W_(-1)). For x ≥ 0, there is only one real solution, which lies on the principal branch. The derivative of the Lambert W function is given by W'(x) = W(x) / [x(1 + W(x))], which is well-defined except at the branch point x = -1/e, where W(x) = -1 and the derivative is unbounded. At x = 0, the derivative of the principal branch is 1.

Experimenting with Branches

You can use the interactive module to explore the two real branches of the Lambert W function.

Select the Input from the dropdown to probe different regions of the domain.
Select the Branch to evaluate either the Principal (W_0) or Lower (W_(-1)) branch.
Observe the plotted value and how it lies on the curve (w e^w, w).

Worked Example

Let's find the value of the Lambert W function on the principal branch for x = 1. We want to solve w e^w = 1. Using a root-finding algorithm like Halley's method:

Initial guess: w_0 = 1. Evaluate f(w_0) = 1 · e^1 - 1 = e - 1 ≈ 1.71828.
Update guess using Halley's formula.
Convergence yields W_0(1) ≈ 0.5671432904097838.

Checking the residual: 0.5671432904097838 · exp(0.5671432904097838) ≈ 1.0. The residual is 0.

Limitations

This module is restricted to real inputs and the two real branches (W_0 and W_(-1)). It does not compute complex branches (W_k for k ∉ {0, -1}) which are required for complex inputs or real inputs below the branch point x < -1/e.

Primary References

For more information, see the NIST Digital Library of Mathematical Functions (DLMF):
DLMF 4.13: Lambert W-Function

Lambert W Real Branches