Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

ProductLog threads elementwise over lists and matrices:

ProductLog can be used with Interval and CenteredInterval objects:

Values of ProductLog at fixed points:

Values at zero:

Values at infinity:

Find a value of x for which the ProductLog[x]=0.5 using FindRoot:

Plot the ProductLog function:

Plot the real part of :

Plot the imaginary part of :

Polar plot with :

ProductLog is defined for all real values from the interval [-,∞):

ProductLog is defined for all complex values:

The two-argument form requires that be an integer and :

The real range:

ProductLog is not an analytic function:

Nor is it meromorphic:

ProductLog is increasing on its real domain:

ProductLog is injective:

ProductLog is not surjective:

ProductLog is neither non-negative nor non-positive:

ProductLog has both singularity and discontinuity in (-∞,-]:

ProductLog is concave on its real domain:

TraditionalForm formatting:

The first derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z:

Derivative of a nested logarithmic function:

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral of ProductLog:

More integrals:

Find the Taylor expansions using Series:

Plots of the first three approximations around :

Expand the two-argument form:

The general term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Find series expansions at branch points and branch cuts:

The series expansion at infinity contains nested logarithms:

ProductLog gives the solution for the following equation:

Expand assuming real variables x and y: