Is there any representation for kth branch of Lambert-$W$ function?

Question

How to represent the infinite branches of Lambert$W$ function? I just tried to solve it graphically in mathematica and got an interesting representation of the function, which is helpful in understanding this function. Please check my solution.

Answer 1

In mathematics, the Lambert-W function, also called the omega function or product logarithm, is a set of functions, namely the branches of the inverse relation of the function $f(z) = ze^{z}$ where $e^{z}$ is the exponential function and z is any complex number. In other words $$z=f^{-1}(ze^{z})=W(ze^{z})$$ Now, Taking equation $ze^{z}=c$, $c\in R$. This imply $z=W_{k}(c)$. where $W_{k}$ for k an integer are countably many branches of the W function. $W_{0}$ being the principal branch. $W_{0}(x)$ is real and positive for $x \in (0,\infty)$ and $W_{0}(x)$,$W_{-1}(x)$ both are real and negative for $x \in (\frac{-1}{e},0)$. and all other $W_{k}(x)$ are complex $\forall x$.

Subsituting, $z=x+iy$, and seperating real and imaginary parts we get $$e^{x}(x\cos{y}-y\sin{y})=c~~\&~~ie^{x}(x\sin{y}+y\cos{y})=0.$$ After further simplification we get $$x=-y\cot{y}~~\&~~x=\log{-\frac{c\sin{y}}{y}}$$

I calculated the point of intersection of this graph for $c=1$. To find out y cordinate of point of intersection.
$$FindRoot[- y*\cot[y] - Log[-1*\sin[y]/y], (y, \frac{3\pi}{2})]$$ this command gives root near $y=\frac{3\pi}{2}$ as it is can be seen in graph that we have root near $y=\frac{3\pi}{2}$ and the output is $(y \rightarrow4.37519)$. putting this output in $-y\cot[y]$ we get $x \rightarrow -1.53391$, Thus, (-1.53391,4.37519) will satisfy $ (x+iy )*e^{-(x+iy)} = 1.$ And from the symmetry of graph (-1.53391,-4.37519) is also a solution.

Input - $$Table[ ProductLog[k, 1], {k, -2, 2}] // N$$ Output- $$ {-2.401 - 10.77 I, -1.533 - 4.375 I, 0.5671, -1.533 + 4.375 I, -2.401 + 10.776 I} $$ Now, we can see the point of intersection exatly matches with Lambert W function. These are the values of $W_{k}(1)$ from $k=-2~to~2$. From here we can atempt to write general expression of $W_{k}$. $$Im(W_{k}(c))=y:-y\cot{y}=\log{-\frac{c\sin{y}}{y}}, \begin{cases}{} 2k\pi <y<(2k+1)\pi~~for~~k\geq0 \\ (2k+1)\pi <y<(2k+2)\pi~~for~~k<0\\ \end{cases} $$ $$Re(W_{k}(c))= -y\cot{y}~or~\log{-\frac{c\sin{y}}{y}}: y ~is ~the~ Im(W_{k}(c))$$ This expression is true for $c<\frac{-1}{e}$ where all roots are imaginary. For $\frac{-1}{e}<c<0$ this expression is still true except for real roots $W_{-1}$ and $W_{0}$.

For $c>0$ only $W_{0}$ is real, thus expression is: $$Im(W_{k}(c))=y:-y\cot{y}=\log{-\frac{c\sin{y}}{y}}, \begin{cases}{} (2k-1)\pi <y<2k\pi~~for~~k>0 \\ (2k)\pi <y<(2k+1)\pi~~for~~k<0\\ \end{cases} $$ $$Re(W_{k}(c))= -y\cot{y}~or~\log{-\frac{c\sin{y}}{y}}: y ~is ~the~ Im(W_{k}(c))$$