Online Lambert W(x) function calculator

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The calculator

The calculator below evaluates the Lambert function W(x) for the variable x, which is defined to be the solution to the equation:

Equivalently, the Lambert function is the inverse of the function:

Simply enter a value for x and click “Calculate W(x)” to find the value of the function.

Click here to see how the Lambert function can be used in order to solve the transcendental equation below:

Analysis of the Lambert function

In order to analyze the Lambert function, it is easier to consider its inverse function f(x) as defined above. First, we note that the domain of this inverse function is clearly (-∞,+∞). In order to find its range, we calculate the 1^st and 2^nd derivatives:

We see that f(x) has an extremum for x=-1:

It follows that the extremum is a global minimum with value -1/e and the maximum of the invserse function is +∞. Its range is therefore [-1/e,∞). As a consequence, the domain of the Lambert function is [-1/e,∞) and the range is (-∞,+∞). From the analysis of the inverse function f(x), we conclude that W(x) satisfies the following relations:

Since the inverse function f(x) has a global minimum, it follows that the Lambert function W(x) has two branches:

• One with domain [-1/e,+∞) and range [-1,+∞).
• One with domain [-1/e,0) and range (-∞,-1].

As a result, the calculator returns two values for x in the interval [-1/e,0), and a single value for x in the interval [0,+∞). Using the calulator, the graph of the Lambert function can be plotted:

Properties of W(x)

From its definition, it follows that W(x) satisfies the functional relations:

Differentiation

Using implicit differentiation and the above functional relations, we can easily calculate the 1^st derivative:

The 2^nd derivative can then be obtained as well:

Integration

In order to find an antiderivative of W(x), we can start with an integration by parts:

Writing the integrand of the remaining integral as

leads to:

Using the expression of the derivative found earlier, we can rewrite the infinitesimal element dx in terms of dW:

This leads to:

Putting everything together, we have shown that:

Uses of the Lambert function

The Lambert function can be used in order to solve some transcendental equations. For example, consider the equation:

We can proceed as follows:

We can make the following change of variable:

This implies:

Finally, inverting the change of variable, we conclude:

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