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Project: BHLectures
Views: 43
Kernel: SageMath 9.3.beta9

Lambert function W0W_0

version()
'SageMath version 9.3.beta9, Release Date: 2021-03-14'
%display latex
F(x) = exp(x)*(x-1) F
x ↦ (x−1)ex\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
Image in a Jupyter notebook
F(-1)
−2 e(−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
Image in a Jupyter notebook

Rescaled Lambert function

def w(x): return RDF(lambert_w(RDF(x/e)) + 1)
w(0)
1.0\renewcommand{\Bold}[1]{\mathbf{#1}}1.0
w(-0.9999999)
0.0004471469422080432\renewcommand{\Bold}[1]{\mathbf{#1}}0.0004471469422080432
w(2)
1.463055513365549\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
Image in a Jupyter notebook