Gold 9

Lambert W function

Gold ID

9

Link

https://sigir21.wmflabs.org/wiki/Lambert_W_function#math.59.52

Formula

$\begin{array}{rl}x& =u{e}^u,\\ \frac{dx}{du}& =(u+1)e^u.\end{array}$

TeX Source

\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Semantic LaTeX

Translation

\begin{align}x &= u \expe^u , \\ \frac{dx}{du} &=(u + 1) \expe^u .\end{align}

Expected (Gold Entry)

\begin{align}x & =\LambertW@{x}\expe^{\LambertW@{x}}, \\ \deriv{x}{\LambertW@{x}} &=(\LambertW@{x} + 1) \expe^{\LambertW@{x}} .\end{align}

Mathematica

Translation

x == u*Exp[u] Divide[d*x,d*u] == (u + 1)*Exp[u]

Expected (Gold Entry)

x == ProductLog[x]*(E)^(ProductLog[x]) D[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]

Maple

Translation

x = u*exp(u); (d*x)/(d*u) = (u + 1)*exp(u)

Expected (Gold Entry)

x = LambertW(x)*exp(u); diff(x, [LambertW(x)$\$$1]) = (LambertW(x) + 1)*exp(LambertW(x))