Lambert W Function

analysis

Definition

Lambert W Function

The Lambert $W$ function is the multivalued inverse function of the map

$$f(w)=w{e}^w.$$

For any $z\in \mathbb{C}$, every branch $W_k$ satisfies

$$W_k(z)e^{W_k(z)}=z.$$

In particular, solving $x{e}^x=a$ gives $x=W(a)$ on an appropriate branch.

Branch Structure

The function is not single-valued on $\mathbb{C}$. Its branches are denoted by $W_k$, where $k\in \mathbb{Z}$. The principal branch is $W_0$.

On the real line:

• $W_0(x)$ is real for $x\ge -e^{-1}$.
• $W_{-1}(x)$ is real for $-e^{-1}\le x<0$.

The point $z=-e^{-1}$ is the real branch point where $W_0$ and $W_{-1}$ meet.

Differentiation

Away from branch points, for $z\ne 0$,

$$W^{\prime }(z)=\frac{W(z)}{z\left(1+W(z)\right)}.$$

Equivalently, if $y=W(z)$, then

$$\frac{dy}{dz}=\frac{1}{e^y(1+y)}.$$

Series Expansion

The principal branch admits the power series

$$W_0(z)=\sum _{n=1}^{\infty }\frac{(-n{)}^{n-1}}{n!}{z}^n$$

with radius of convergence $e^{-1}$.

Use in Equation Solving

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

$$x{e}^x=a$$

into a direct closed-form solution using $W$.

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