Lambert W function

Builds on Exponential function Inverse function

Definition

Let $f \colon \R \to \R$ be $f(w) = w e^w$ , where $e^w$ is the exponential function.

The principal branch $W_0 \colon [-1/e, \infty) \to [-1, \infty)$ is the inverse of the restriction of $f$ to $[-1, \infty)$ .
The lower branch $W_{-1} \colon [-1/e, 0) \to (-\infty, -1]$ is the inverse of the restriction of $f$ to $(-\infty, -1]$ .

Thus $W(x)\, e^{W(x)} = x$ for both branches $W$ on their domains.

Remarks

The two restrictions are bijections because $f'(w) = (1 + w) e^w$ is positive for $w > -1$ and negative for $w < -1$ , $f(-1) = -1/e$ , $f(w) \to 0^-$ as $w \to -\infty$ , and $f(w) \to \infty$ as $w \to \infty$ .

Example: $W_0(e) = 1$ , because $1 \cdot e^1 = e$ and $1 \ge -1$ . Also $W_0(0) = 0$ , $W_0(-1/e) = W_{-1}(-1/e) = -1$ , and $W_0(1) = \Omega \approx 0.5671$ , the omega constant.

Non-example: $x = -1$ is in the domain of neither branch, because $w e^w \ge -1/e > -1$ for every real $w$ .

Over $\C$ , the equation $w e^w = z$ has infinitely many solutions for $z \ne 0$ , and they are grouped into branches $W_k$ , $k \in \Z$ , which extend $W_0$ and $W_{-1}$ . Implicit differentiation gives $W'(x) = \frac{W(x)}{x\,(1 + W(x))}$ for $x \ne 0, -1/e$ .

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Tag	`R1XGWTVY3N`
URI	`m2://R1XGWTVY3N`
Kind	definition
Writing	W0 stub
MSC	33E20, 33B10
Defines	Lambert W function
Uses	Exponential function, Inverse function

Lambert W function

Definition

Remarks

Reading path

Further reading

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