Does this equation have any solution?

[Petrushka]

x\multsp =\multsp {{e}^x}

I'm aware there's no real solution, but does any complex solution exist?

 

[dextercioby]

Yes,it has 2 distinct solutions:

x_{1}=-W(-1)

x_{2}=-W(-1,-1)

,where W is the Lambert function.

Daniel.

 

[Petrushka]

I see. Having read a bit about the Lambert function on Http://www.mathworld.com it doesn't make much sense to me.

Is the following value:

x_{1}=-W(-1)

...a complex number, or something entirely different?

 

[dextercioby]

You mean a # of nonzero imaginary part...?Yes it is.As u said and as a graph and elementary calculus would show,for real #-s the equation does not have solution.

Daniel.