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The principal branch of the Lambert W function is the inverse function of y=x ex for x -1. If one extends the range of definition for the latter function to arguments smaller than minus one, the function is not monotonic anymore. Thus, one can divide the range of definition in two sets (x≤-1 and x -1), where different inverse functions are valid. Both inverse functions agree for x=-1. In our case, x=-b≤-1 and thus the principal branch of the Lambert W function is not the inverse function. Therefore, the trivial identity A=-b+b=0 is only fulfilled in the limit b→-1.
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The principal branch of the Lambert W function is the inverse function of y=x ex for x -1. If one extends the range of definition for the latter function to arguments smaller than minus one, the function is not monotonic anymore. Thus, one can divide the range of definition in two sets (x≤-1 and x -1), where different inverse functions are valid. Both inverse functions agree for x=-1. In our case, x=-b≤-1 and thus the principal branch of the Lambert W function is not the inverse function. Therefore, the trivial identity A=-b+b=0 is only fulfilled in the limit b→-1.
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