n | 1 | x 2 | sqrt(x) 3 | x^(1/3)

n | 2 | 3 |

Re(n)>0, x = 0

n = 1, x = 0

n = 2, x = 0

n = 3, x = 0

n = 4, x = 0

n = 5, x = 0

{x element R : (x!=0 and 1/n element Z) or (0=0 and n>0) or x>0}

d/dx(x^(1/n)) = x^(1/n - 1)/n

integral x^(1/n)dx = (n x^(1/n + 1))/(n + 1) + constant

x^(1/n) = sum_(ν=0)^∞ binomial(1/n, ν) (-1 + x)^ν

x^(1/n) = sum_(ν=0)^∞ c_ν (-1 + n)^ν for (c_0 = x and c_1 = -x log(x) and ν c_ν + (2 + 2 ν + log(x)) c_(1 + ν) + (2 + ν) c_(2 + ν) = 0)

(1 + z)^a = ( integral_(-i ∞ + γ)^(i ∞ + γ) (Γ(s) Γ(-a - s))/z^s ds)/((2 π i) Γ(-a)) for (0<γ<-Re(a) and abs(arg(z))<π)