Does this equation have any solution?

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SUMMARY

The equation \( x = e^x \) has no real solutions but possesses two distinct complex solutions: \( x_{1} = -W(-1) \) and \( x_{2} = -W(-1, -1) \), where \( W \) denotes the Lambert function. The Lambert function is crucial for understanding these solutions, as it provides the necessary framework for complex analysis in this context. The discussion confirms that \( x_{1} \) is indeed a complex number with a nonzero imaginary part, reinforcing the conclusion that real numbers do not satisfy the equation.

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

I'm aware there's no real solution, but does any complex solution exist?
 
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Yes,it has 2 distinct solutions:

x_{1}=-W(-1)

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

,where W is the Lambert function.

Daniel.
 
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?
 
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.
 

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