Tommy's Gaussian method.

[tommy1729]

I conjecture

all t(s) with t(s) = ( 1 + u(s)) /2 and phi(s) = exp(t(s-1) exp(...))  will give analytic tetration solutions if 

( sufficient conditions )

1) u(s) = integral f(t) erf(s t) dt from 0+ to +oo.

2) u(s) and f(s) are analytic 

3) integral f(t) dt from 0+ to +oo = 1 and integral f(t)^2 dt from 0+ to +oo is convergeant.

4) f(t) >= 0.

regards

tommy1729