(x - 1)!

(no roots exist)

1/x - gamma + 1/12 (6 gamma ^2 + π^2) x + 1/6 x^2 (- gamma ^3 - ( gamma π^2)/2 + polygamma(2, 1)) + 1/24 x^3 ( gamma ^4 + gamma ^2 π^2 + (3 π^4)/20 - 4 gamma polygamma(2, 1)) + O(x^4) (Laurent series)

e^(-x) x^x (sqrt(2 π) sqrt(1/x) + 1/6 sqrt(π/2) (1/x)^(3/2) + 1/144 sqrt(π/2) (1/x)^(5/2) - (139 sqrt(π/2) (1/x)^(7/2))/25920 - (571 sqrt(π/2) (1/x)^(9/2))/1244160 + O((1/x)^(11/2)))

d/dx(Γ(x)) = Γ(x) polygamma(0, x)

Γ(x + 1)≈sqrt(2 π) e^(-x) x^(x + 1/2)

Γ(x)≈sqrt(2 π) e^(-x) sqrt(1/x) (x + 1/(12 x - 1/(10 x)))^x

Γ(x)≈sqrt(2 π) e^(-x) sqrt(1/x) (x sqrt(1/(810 x^6) + x sinh(1/x)))^x

(for x sufficiently far from the negative real line)

Γ(x) = G(1 + x)/G(x)

Γ(x) = e^(-logG(x) + logG(1 + x))

Γ(x) = (-1 + x)!

Γ(x) = ( sum_(k=0)^∞ (x^k Γ^(k)(1))/(k!))/x for abs(x)<1

Γ(x) = 1/( sum_(k=1)^∞ x^k c_k) for (c_1 = 1 and c_2 = 1 and c_k = ( gamma c_(-1 + k) + sum_(j=1)^(-2 + k)-(-1)^(j + k) c_j ζ(-j + k))/(-1 + k))

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

Γ(x) = integral_0^1 log^(-1 + x)(1/t) dt for Re(x)>0

Γ(x) = integral_0^∞ e^(-t) t^(-1 + x) dt for Re(x)>0

Γ(x) = exp( integral_0^1 (-1 + x - x ξ + ξ^x)/((-1 + ξ) log(ξ)) dξ) for Re(x)>0

Γ(x) = (-1 + x) Γ(-1 + x)

Γ(x) = Γ(1 + x)/x

Γ(x) = Γ(n + x)/(x)_n