Lambert W Function Calculator
Use this tool to compute real values of the Lambert W function, defined implicitly by:
Domain note: real solutions exist only for x ≥ -1/e on W₀, and for -1/e ≤ x < 0 on W₋₁.
What is the Lambert W function?
The Lambert W function is the inverse of the expression f(w) = w ew. In other words, if you have an equation shaped like:
then the solution for w is:
It appears in many places where the unknown variable is both inside an exponential and multiplied outside of it.
How to use this calculator
- Enter the value of x in the input box.
- Select the branch:
- W₀ (principal branch): common default branch
- W₋₁ (lower branch): useful for specific negative-range problems
- Choose output precision and click Calculate.
- The result includes the computed value, a check of
W(x)eW(x), and residual error.
Branch behavior and domain restrictions
Principal branch W₀(x)
The principal real branch is defined for x ≥ -1/e. It returns values from -1 upward and is the branch most people use first.
Lower branch W₋₁(x)
The lower real branch is defined only on -1/e ≤ x < 0. On this branch, values are less than or equal to -1, often much smaller for x near 0 from below.
Why this function matters
The Lambert W function helps solve equations that are otherwise awkward to isolate. Typical examples include:
- Population or growth models with delayed/implicit exponential terms
- RC/RL circuit equations and diode-related formulas
- Queueing and performance equations in computer systems
- Closed-form rearrangements in physics and applied mathematics
- Certain continuous compounding and optimization relationships in finance
Worked examples
Example 1: Solve w ew = 1
This gives w = W₀(1), also called the omega constant, approximately 0.567143....
Example 2: x = -0.1
This input has two real outputs:
- On W₀: a value between -1 and 0
- On W₋₁: a value less than -1
Use the branch selector to switch and compare both real roots.
Numerical method used in this page
This calculator uses an iterative solver (Halley-style updates with safeguards) to compute high-accuracy real values. It starts from branch-aware initial guesses and then refines quickly until convergence.
At the end, it reports a residual error:
A tiny residual means the result is numerically consistent with the defining equation.
FAQ
Can this calculator return complex values?
No. This version is focused on real-valued branches W₀ and W₋₁ only.
Why do I sometimes see a domain error?
Because real Lambert W is not defined everywhere on every branch. Check whether your x value lies inside the selected branch's valid interval.
How accurate are the outputs?
For well-conditioned real inputs in valid domains, the results are typically very accurate (near machine precision), and the residual check is shown so you can verify quality directly.