Lambert_W0.ipynb

Kernel: SageMath 9.3.beta9

Lambert function $W_0$

version()

'SageMath version 9.3.beta9, Release Date: 2021-03-14'

%display latex

F(x) = exp(x)*(x-1)
F

$$\renewcommand{\Bold}[1]{\mathbf{#1}}x \ {\mapsto}\ {\left(x - 1\right)} e^{x}$$

graph = plot(F(x), (x,0, 2), axes_labels=[r'$x$', r'$F(x)$'], thickness=2, gridlines=True)
graph.save('max_X2mT2.pdf')
graph

F(-1)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}-2 \, e^{\left(-1\right)}$$

graph = plot(lambert_w(x), (x,-1/e, 2.7), axes_labels=[r'$y$', r'$W_0(y)$'], 
              thickness=2, gridlines=True)
graph

Rescaled Lambert function

def w(x):
    return RDF(lambert_w(RDF(x/e)) + 1)

w(0)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}1.0$$

w(-0.9999999)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}0.0004471469422080432$$

w(2)

$$\renewcommand{\Bold}[1]{\mathbf{#1}}1.463055513365549$$

graph = plot(w, (x, -1, 5), aspect_ratio=1, axes_labels=[r'$x$', r'$\tilde{W}_0(x)$'], 
             thickness=2, gridlines=True)
graph.save('max_lambert_rescaled.pdf')
graph