How do you solve for $x in RR$ the equation $x! = e^x$ ?

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Answer 1 · 2017-03-05T06:30:05

I got $x~=5.290316$

I used the extension $x! = Gamma (x+1)$ iteratively in the range suggested by Cesareo R. to come up with 5.290316

Obviously, equation is also satisfied for $x=0$ by definition.

Answer 2 · 2017-03-05T15:50:55

Agree $x=5.290316...$ is a solution using the Gamma function, but by the nature of the question I assumed we were looking for integer solutions.

As Gamma grows faster then the exponential I would assume this is the only solution.

graph{(y-x!)(y-e^x)=0 [-10, 10, -50, 300]}

I'll post an asymptotic solution if I get time.

How do you solve for x in RRx in RR the equation x! = e^xx! = e^x ?