Lambert W Function

analysis

Definition

Lambert W Function

The Lambert function is the multivalued inverse function of the map

For any , every branch satisfies

In particular, solving gives on an appropriate branch.

Branch Structure

The function is not single-valued on . Its branches are denoted by , where . The principal branch is .

On the real line:

• is real for .
• is real for .

The point is the real branch point where and meet.

Differentiation

Away from branch points, for ,

Equivalently, if , then

Series Expansion

The principal branch admits the power series

with radius of convergence .

Use in Equation Solving

The Lambert function is useful when the unknown appears both as a factor and in an exponent. Typical reductions transform equations of the form

into a direct closed-form solution using .

This often appears after algebraic rearrangement of equations involving exponential functions and logarithms.