`lambertw`

Lambert W function, the inverse of w*exp(w).

Category: Mathematics > Mathematical Functions

Syntax

W = lambertw(x)

Principal branch W0(x).

W = lambertw(k, x)

Branch number k (an integer).

Description

The equation w*exp(w) = x has infinitely many complex solutions. lambertw picks one per branch. For real x >= -1/e the principal branch is real, and for -1/e <= x < 0 the branch k = -1 gives a second real solution that is <= -1.

At the branch point x = -1/e both real branches meet at w = -1. The function solves equations where an unknown appears both as a factor and inside an exponential.

Input arguments

k: Branch index, an integer. 0 is the principal branch; -1 is the other real branch.
x: Argument, real or complex, scalar or array.

Output arguments

W: Solution w of w*exp(w) = x on the chosen branch.

Example

Both real branches

lambertw(1)
lambertw(0, -0.2)
lambertw(-1, -0.2)
w = lambertw(1);
w*exp(w)

Output

ans = 0.5671
ans = -0.2592
ans = -0.2592
ans = 1

lambertw