# Does this equation have any solution?

1. Mar 9, 2005

### Petrushka

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

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

2. Mar 9, 2005

### dextercioby

Yes,it has 2 distinct solutions:

$$x_{1}=-W(-1)$$

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

,where W is the Lambert function.

Daniel.

3. Mar 9, 2005

### 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?

4. Mar 9, 2005

### 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.