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While solving $e^z=z$, I realise there is no real solution and want to find a complex solution, if any.

I searched around the internet only to find functions describing approximations that might be sufficiently close, but I can't find a way to know the exact answer.

My question is :

DERIVE all solutions to $e^z=z$ (and prove if there are none) :-)

I'm NOT interested in the answer, I am interested in the solution

anonymous
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1 Answers1

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HINT: defining $$f(x)=e^{x}-x$$ and then we have $$f'(x)=e^{x}-1$$ and $$f''(x)=e^{x}>0$$ and the solution is given by $$x=-{\rm W} \left(-1\right)$$ with the LambertW-function. and the complex solution is given by $$x\approx 0.3181315052- 1.337235701\,i$$

Dr. Sonnhard Graubner
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