Equation between the two branches of the lambert w function

Question

Is there an equation connecting the two branches $W_0(y)$ and $W_{-1}(y)$ of the Lambert W function for $y \in (-\tfrac 1e,0)$?

For example the two square roots $r_1(y)$ and $r_2(y)$ of the equation $x^2=y$ fulfill the equation $r_1(y)=-r_2(y)$. So if one has computed one root, he already knows the second one by taking the negative of the computed root. It is also possible to calculate $W_0(y)$ by knowing $W_{-1}(y)$ and vice versa?

Note: I reasked this question on mathoverflow. Because I read that questions shall not be migrated when they are older than 60 days I didn't asked for migration. I hope that's okay...

Answer 1

$\require{begingroup} \begingroup$

$\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$

Using the two roots of the quadratic equation analogy, if we know two values, the sum $a_1$ and the product $a_0$ of the roots \begin{align} x_0+x_1&=a_1 ,\\ x_0x_1&=a_0 , \end{align}

we know how to express the values of both roots in terms of these two values, $a_1$ and $a_0$.

The case with the two values $\Wp(x),\ \Wm(x)$ for $x\in(-\tfrac1\e,0)$ is even more simple and interesting: we indeed can find both, if we just know one value, either the fraction

\begin{align} \frac{\Wp(x)}{\Wm(x)}&=a ,\quad a\in(0,1) \end{align}

or the difference

\begin{align} \Wm(x)-\Wp(x)&=b ,\quad b\le0 . \end{align}

Given $b$, we have $a$ as

\begin{align} a&=\exp(b) . \end{align}

And if we know $a$, than

\begin{align} \Wp(x)&=\frac{a\ln a}{1-a} ,\\ \Wm(x)&=\frac{\ln a}{1-a} , \end{align}

which is Parametric representation of the real branches $\operatorname{W_{0}},\operatorname{W_{-1}}$ of the Lambert W function.

$\endgroup$

Answer 2

Alexandre Eremenko provided a good answer on Mathoverflow. See https://mathoverflow.net/a/195932/56668