todo: write down the above with an expanded $\log$ to third order

Because of this, the local minimum of $x^x$ is that of $x\log(x)$, namely $\frac{1}{\mathrm e}\approx 0.37$, and then see Secretary problem (Wikipedia)

Furthermore

with the Pochhammer symbol $(1-x)_n:=\prod_{k=0}^{n-1} (k-x)=\prod_{k=1}^n (k-(1+x))$.

From this

and all of this appears more naturally via

$x^x \equiv (1-t)^{1-t} = (1-t)\dfrac{1}{(1-t)^t}$

$=1-t+t^2-\frac{1}{2}t^3+\frac{1}{3}t^4-\frac{1}{12}t^5+\frac{3}{40}t^6+\dots$

as Binomials relate to Gammas and Betas.

For moral talk regarding the rewriting $x=1-t$, see Zeta functions.

$P_n(x) := (1-x) \, x^n \prod_{k=0}^{n-1} \, (x+k)$

this must come from the Expression at the beginning, but I don't know how I arrived there.
Maybe there I switched from $x$ to $1-x$ or $\int_0^1$ to $\int_1^0$.

$\int_0^1 \, x^x \, dx = \sum_{n=0}^\infty \frac{1}{n!} \int_0^1 \, P_n(x) \, dx = \frac{1}{2} + \frac{1}{12} + \frac{1}{24} + \frac{131}{5040} + \frac{1093}{60480} + \dots$

Wikipedia: Elementary function, Secretary problem

gx_fx

Euler beta function