TheLambertWFunctionIsTheNewtonFlowOfTheExponentialFunction

The Lambert W function, denoted by $W(z)$, is defined as the inverse function of $f(z) = ze^z$. In other words, if $w = W(z)$, then we have $z = w e^w$.

The continuous Newton's method is a technique for finding the roots of a function by following the flow of its gradient vector field. Specifically, given a function $f(x)$ and an initial point $x_0$, the continuous Newton's method generates a solution curve $x(t)$ that satisfies the differential equation:

$$\frac{dx}{dt} = -\nabla f(x)$$

where $\nabla f(x)$ is the gradient vector of $f$ at $x$.

In the case of the function $f(z) = ze^z$, the gradient vector is given by:

$$\nabla f(z) = (1+z)e^z$$

Therefore, the differential equation for the continuous Newton's method is:

$$\frac{dz}{dt} = -(1+z)e^z$$

which is exactly the derivative of the Lambert W function. Therefore, the solution curve generated by the continuous Newton's method starting at a point $z_0$ is given by:

$$z(t) = W(z_0 e^{-t}).$$

In other words, the Lambert W function is the flow of the exponential function under the continuous Newton's method.