For what $z$ do $\Gamma(x)$ and $x^ze^{-x}$ share a tangent line at exactly one point?

Question

The motivation for this question is that $\Gamma(x)$ is defined as $\int_0^{\infty}t^{x-1}e^{-t}dt$. When plotting both $y=\Gamma(x)$ and $y=x^ze^{-x}$ for different values of $z,$ the two functions "kiss," i.e. they share a tangent line at exactly one point, at a value around $z=2.88518212.$ What is this number exactly? Does it have a closed form? What is the slope of the tangent line at said "kissing point?"

Answer 1

The two equations to be solved are $$\Gamma (x)-x^z\, e^{-x} =0 \tag 1$$ $$e^{-x} (x-z) x^{z-1}+\Gamma (x) \,\psi (x) =0 \tag 2$$

$$(1)\qquad \implies\qquad \color{red}{z(x)=\frac{\log \left( \Gamma (x)\right)+x}{\log (x)}}\tag 3$$ Plug in $(2)$ to obtain $$\Gamma (x) \left(\psi (x)-\frac{\log \left(e^x \Gamma (x)\right)}{x \log (x)}+1\right)=0\tag 4$$

Finally, the equation to be solved is $$x \log (x) (1+\psi (x))-(x+\log (\Gamma (x)))=0\tag 5$$ The solution of $(5)$ is close to $x_0=2$. The first iterate of Newton method gives $$x_1=\frac{2 \left(3+\left(\pi ^2-6\right) \log (2)\right)}{\left(\pi ^2-3 \gamma \right) \log (2)}$$ which, numerically, gives $$z(x_1)=\color{red}{2.8851821}313\cdots$$ to be compared to the exact solution $$z= \color{red}{2.8851821225\cdots}$$

Trying to find a good approximation of the solution of $(5)$ in terms of basic constants $$x\sim -\frac{1+4 \sqrt{2}+3 \sqrt{3}-4 e+\pi (3+\pi )-3 \log (3)+3\log (2)}{3-3 \sqrt{2}-8 e+2 (\pi -2) \pi +6\log (2)+2\log(3)}$$ whose error is $4.13\times 10^{-19}$.

This value gives $$z=\color{red}{2.885182122499993175519796224373851235}306\cdots$$ to be compared to the exact solution $$z=\color{red}{2.885182122499993175519796224373851235141\cdots}$$