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if $x^y =y^x$ then what is all possible solution on $\mathbb{C}$

I was playing with graphing calculator and tried the equation $x^y =y^x$ and I was expecting another set of solution other that the trivial solution $x=y$ because $2^4 =4^2 $ and when I graphed the equation I got enter image description here

but I have no Idea what is the formula of the green curve I could just proof that the intersection between the $x=y$ and the other set of solutions is $(e,e)$ because $e^a >a^e$ and I am curious if there is another set of solutions on $\mathbb{C}$ ( I don't know how to graph complex function and i don't know if it is even possible to graph function of several complex variables also I haven't study complex analysis yet )

pie
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  • For real $x,y$ this has been solved here already, see here. For complex numbers, we first need to define what complex powers are - see the discussion here. – Dietrich Burde Sep 19 '23 at 14:16
  • @BenGrossmann how did you search for duplicates ? when I wrote the question title this question didn't appear to me – pie Sep 19 '23 at 14:18
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    @pie Well I knew that this question exists because I wrote the top answer – Ben Grossmann Sep 19 '23 at 14:19
  • @BenGrossmann oh! I didn't notice that – pie Sep 19 '23 at 14:20
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    Actually the second-most upvoted answer in the link @BenGrossmann has given you gives the solution explicitly in terms of the Lambert W function, which is well defined for complex values I believe if memory serves me right but you have to be careful about branch cuts and whatnot – Riemann'sPointyNose Sep 19 '23 at 14:21
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    In fact, if you plug in ${x=i}$ to the formula given in the link, using whatever branch Wolfram Alpha uses you get ${y = -i}$. I.e. ${(-i)^i = i^{(-i)}}$, which is true with whatever branch Wolfram uses – Riemann'sPointyNose Sep 19 '23 at 14:28
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    This equation can be written as $x^{1/x}=y^{1/y}.$ The function $f(w)=w^{1/w}$ increases for $0\leq w<e,$ has a local maximum at $w=e$ then decreases, with $f(w)\to1=f(1)$ as $w\to\infty.$ So for any $1<x<e,$ by the intermediate value theorem, there is a $y>e$ such that $f(x)=f(y).$ But the formula for $y$ in terms of $x,$ if it exists, probably requires a special function, like the Lambert W function. – Thomas Andrews Sep 19 '23 at 14:41

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