Lambert W Functions
Lambert’s \(W\) functions, \(W(x)\), are defined to be solutions of the
equation \(W(x) \exp(W(x)) = x\). This function has multiple branches
for \(x < 0\); however, it has only two real-valued branches. We define
\(W_0(x)\) to be the principal branch, where \(W > -1\) for \(x < 0\), and
\(W_{-1}(x)\) to be the other real branch, where \(W < -1\) for \(x < 0\).
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gsl_sf_lambert_W0(x)
This routine computes the principal branch of the Lambert \(W\) function,
\(W_0(x)\).
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gsl_sf_lambert_Wm1(x)
This routine computes the secondary real-valued branch of the Lambert
\(W\) function, \(W_{-1}(x)\).