Literature for x^x and Sophomore's Dream

[tommy1729]

Multiplicative integral and discrete multiplicative integral for this function are known to be

\(
e^{-\frac{1}{4}x^2+\frac{1}{2}x^2\ln x}
\)

and

\(
e^{\frac{z-z^2}{2}+\frac z2 \ln (2\pi)-\psi^{(-2)}(z)}
\)

respectively.

Conventional integral and discrete integral are not known.

just like log(x)/log(x+1) , dx / log(x) , exp(x^3) , exp(x^2) , exp(x)/x , sin(x)/x , cos(x)/x , arctan(x)/x , x^x has no conventional integral in terms of elementary functions.

however you may want to look at lagrange inversion , Barnes G , Lambert W and perhaps Meijer G since they relate strongly.

AFAIK Sophomore's dream has never been extended , but lagrange inversion is strongly related.

in fact i believe the extention of Sophomore's dream is more important and relevant to tetration than x^x is !!

as for equating sums to integrals , euler and maclaurin ( and perhaps newton ) have intresting formula's , yet i dont think they help much at first sight ...
( i think lagrange inversion is more important at first sight )

regards

tommy1729