e^x !

Γ(e^x + 1)

(no roots exist)

1 + (1 - gamma ) x + 1/12 (6 - 18 gamma + 6 gamma ^2 + π^2) x^2 + x^3 ( gamma ^2 - gamma ^3/6 - 1/12 gamma (14 + π^2) + 1/6 (-1 + π^2 + polygamma(2, 2))) + (x^4 (-1140 - 600 gamma ^3 + 60 gamma ^4 + 250 π^2 + 9 π^4 + 60 gamma ^2 (25 + π^2) + 600 polygamma(2, 2) - 60 gamma (7 + 5 π^2 + 4 polygamma(2, 2))))/1440 + O(x^5) (Taylor series)

sqrt(2 π) exp(-e^x + 1/(12 (e^x + 1)) - 1/(360 (e^x + 1)^3) + 1/(1260 (e^x + 1)^5) - 1/(1680 (e^x + 1)^7) + 1/(1188 (e^x + 1)^9) - 691/(360360 (e^x + 1)^11) + 1/(156 (e^x + 1)^13) - 3617/(122400 (e^x + 1)^15) + 43867/(244188 (e^x + 1)^17) - 174611/(125400 (e^x + 1)^19) + 77683/(5796 (e^x + 1)^21) - 236364091/(1506960 (e^x + 1)^23) + 657931/(300 (e^x + 1)^25) - 3392780147/(93960 (e^x + 1)^27) + (e^x + 1/2) log(e^x + 1) - 1)

d/dx(exp(x)!) = e^x Γ(1 + e^x) polygamma(0, 1 + e^x)

min{exp(x)!}≈0.88560 at x≈-0.77299

lim_(x->-∞) e^x ! = 1

exp(x)! = Γ(1 + exp(x))

exp(x)! = (1)_exp(x)

exp(x)! = (-1 + exp(x))!! exp(x)!!

integral_0^(i π) e^x ! dx≈50....

exp(x)! = sum_(k=0)^∞ ((e^x - n_0)^k Γ^(k)(1 + n_0))/(k!) for ((n_0 not element Z or n_0>=0) and e^x->n_0)

exp(x)!∝exp(-e^x + sum_(k=0)^∞ ((e^x)^(-1 - 2 k) B_(2 + 2 k))/(2 + 6 k + 4 k^2)) (e^x)^(1/2 + e^x) sqrt(2 π) for (abs(arg(e^x))<π and e^Re(x)->∞)

exp(x)!∝e^(-e^x) (e^x)^(1/2 + e^x) sqrt(2 π) + e^(-e^x) (e^x)^(1/2 + e^x) sqrt(2 π) sum_(k=1)^∞ sum_(j=1)^(2 k) ((-1)^j 2^(-j - k) (e^x)^(-k) D_(2 (j + k), j))/((j + k)!) for ((abs(arg(e^x))<π and e^Re(x)->∞ and D_(m, j) = (-1 + m) ((-2 + m) D_(-3 + m, -1 + j) + D_(-1 + m, j)) and D_(0, 0) = 1 and D_(m, 1) = (-1 + m)! and D_(m, j) = 0) for m<=3 j - 1)

exp(x)! = integral_0^1 log^(e^x)(1/t) dt for Re(e^x)>-1

exp(x)! = integral_0^∞ e^(-t) t^(e^x) dt for (e^x element Z and e^x>=0)

exp(x)! = integral_1^∞ e^(-t) t^(e^x) dt + sum_(k=0)^∞ (-1)^k/((1 + e^x + k) k!)

exp(x)! = integral_0^∞ t^(e^x) (e^(-t) - sum_(k=0)^m (-t)^k/(k!)) dt for (m element Z and m>0 and -1