# ProductLog

*new to Analytica 5.0*

## ProductLog(z)

Returns the value *x* that solves

- [math]\displaystyle{ z = x * Exp(x) }[/math].

where Exp is the exponential function.

This is also known as the Lambert W function, often denoted as [math]\displaystyle{ W_0(z) }[/math] in mathematical publications. It appears in the analytic solution of many equations involving exponentials and non-exponentials in the same equation.

### Real-valued z

The function is real-valued for a real-valued parameter z ≥ -exp(-1). When «z» is real but less than -exp(-1), it returns NaN when EnableComplexNumbers is 0.

The solution is unique when z ≥ 0 or z = -exp(-1). There are two solutions when -exp(-1) < z < 0, in which case ProductLog returns the main branch, whose value is greater than -1. The secondary branch, whose values are less that -1 in the -exp(-1) < z < 0 interval, is not available from this function.

### Complex numbers

When «z» is a complex number, or z < -exp(-1), the result is a complex number. To obtain the complex result when the parameter is real-valued, you need to set the EnableComplexNumbers system variable to 1.

## History

Introduced in Analytica 5.0.

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