James Knight Wrote:w = ln (ssrt (e^x))This is one of the well known (see wikipedia) formulas re the relation of the functions:
\( x=W(y) \) defined by \( xe^x=y \), \( x>-1 \)
\( x=h(y) \) defined by \( x^{1/x}=y \), \( 0<x<e \)
\( x=\text{ssqrt}(y) \) defined by \( x^x=y \), \( x>1/e \)
(The range conditions are necessary as neither of the functions is injective.)
Well known is:
\( W(x)=\ln(\text{ssqrt}(e^x)) \), \( \text{ssqrt}(x)=e^{W(\ln(z))} \)
\( h(x)=\frac{W(-\ln(x))}{-\ln(x)} \), \( W(x)=xh(e^{-x}) \)
Perhaps not so often mentioned is the derived formula:
\( h(x)=\frac{W(-\ln(x))}{-\ln(x)}=\frac{\ln(\text{ssqrt}(1/x))}{-\ln(x)} \)
However the Lambert W function was not that revolutionary as you describe it. That the function \( xe^x \) is injective for \( x>-1 \) (it has a minimum at -1 as you can easily derive from \( 0=(xe^x)'=e^x+xe^x=e^x(x+1) \)) and its range is \( -1/e\dots \infty \), so there *must* be exactly one solution for values in the range! However this solution is not expressible in terms of \( +,-,*,/,\exp,\ln,\sin,\cos \) (though the proof such a non-expressability statement should be quite difficult). So its more about giving a name to this solution, i.e. \( W \). And it turns out that with this additional function the solution of many other equations can be expressed (for example \( ae^x+be^{cx}=d \) as an exercise).