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## 7.23 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*. The Lambert functions are
declared in the header file ‘`gsl_sf_lambert.h`’.

__Function:__double**gsl_sf_lambert_W0***(double*`x`)__Function:__int**gsl_sf_lambert_W0_e***(double*`x`, gsl_sf_result *`result`)These compute the principal branch of the Lambert W function,

*W_0(x)*.

__Function:__double**gsl_sf_lambert_Wm1***(double*`x`)__Function:__int**gsl_sf_lambert_Wm1_e***(double*`x`, gsl_sf_result *`result`)These compute the secondary real-valued branch of the Lambert W function,

*W_{-1}(x)*.