What is too complicated about $y=x^{xy}$?

Question

While playing with Desmos today, I typed the equation $y=x^{xy}$ and the graph came out to be

I clicked the Learn More option given near my equation and Desmos said: Sometimes the calculator detects that an equation is too complicated to plot perfectly in a reasonable amount of time. When this happens, the equation is plotted at lower resolution.

What is the complication?

Answer 1

The solution is quite simple in terms of Lambert function $$y=x^{xy}\implies y=-\frac{W(-x \log (x))}{x \log (x)}$$ In order to stay in the real domain, you need that $$x \log(x) \geq \frac 1e \implies x \lt \frac{1}{e W\left(\frac{1}{e}\right)}=1.32110\cdots$$

We also have $$y'=\frac{ e^{-2 W(-x \log (x))}}{W(-x \log (x))+1}(\log (x)+1)$$ which cancels at $x=\frac 1e$ and at this point $y=e W\left(\frac{1}{e}\right)=0.756945\cdots$

I think that we have a good description of your function.

Answer 2

We have

$$\begin{align}&y=(x^x)^y\\ \implies &y^{\frac 1y}=x^x\\ \implies &y^{-1}\ln y=x\ln x\\ \implies &-y^{-1}\ln y^{-1}=x\ln x\\ \implies &y^{-1}\ln y^{-1}=-x\ln x\\ \implies &\ln y^{-1} e^{\ln {y^{-1}}}=-x\ln x\\ \implies &W\left(\ln y^{-1} e^{\ln {y^{-1}}}\right)=W\left(-x\ln x\right)\\ \implies &\ln y^{-1}=W\left(-x\ln x \right)\\ \implies &y^{-1}=e^{W\left(-x\ln x\right)}\\ \implies &y=e^{-W\left(-x\ln x\right)}\end{align}$$

This implies, your function is a non-elementary function and can be written as

$$f(x)=e^{-W\left(-x\ln x\right)}.$$

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Then, note that $W(x)$ is real for only $x≥-\frac 1e$.

This means , we have

$$\begin{align}-x&\ln x≥-\frac 1e,\thinspace x>0\\ \implies &x\ln x≤\frac 1e \\ \implies &\ln x e^{\ln x}≤\frac 1e\\ \implies &W\left(\ln xe^{\ln x}\right)≤W\left(\frac 1e\right)\\ \implies &\ln x≤W\left(\frac 1e\right)\\ \implies &0<x≤e^{W\left(1/e\right)}≈1.3211\end{align}$$.