section_path: Non-integer powers of complex numbers/n th roots of a complex number
section_name: n th roots of a complex number
paragraph_text: n th roots of a complex number
  
  Every nonzero complex number z may be written in polar form as
  
  where ρ {\displaystyle \rho } is the absolute value of z, and θ {\displaystyle \theta } is its argument. The argument is defined up to an integer multiple of 2 π; this means that, if θ {\displaystyle \theta } is the argument of a complex number, then θ + 2 k π {\displaystyle \theta +2k\pi } is also an argument of the same complex number for every integer k {\displaystyle k}.
  
  The polar form of the product of two complex numbers is obtained by multiplying the absolute values and adding the arguments. It follows that the polar form of an n th root of a complex number can be obtained by taking the n th root of the absolute value and dividing its argument by n:
  
  If 2 π {\displaystyle 2\pi } is added to θ {\displaystyle \theta }, the complex number is not changed, but this adds 2 i π / n {\displaystyle 2i\pi /n} to the argument of the n th root, and provides a new n th root. This can be done n times, and provides the n n th roots of the complex number.
  
  It is usual to choose one of the n n th root as the principal root. The common choice is to choose the n th root for which − π < θ ≤ π, {\displaystyle -\pi <\theta \leq \pi,} that is, the n th root that has the largest real part, and, if there are two, the one with positive imaginary part. This makes the principal n th root a continuous function in the whole complex plane, except for negative real values of the radicand. This function equals the usual n th root for positive real radicands. For negative real radicands, and odd exponents, the principal n th root is not real, although the usual n th root is real. Analytic continuation shows that the principal n th root is the unique complex differentiable function that extends the usual n th root to the complex plane without the nonpositive real numbers.
  
  If the complex number is moved around zero by increasing its argument, after an increment of 2 π, {\displaystyle 2\pi,} the complex number comes back to its initial position, and its n th roots are permuted circularly (they are multiplied by e 2 i π / n e^{2i\pi /n}). This shows that it is not possible to define a n th root function that is continuous in the whole complex plane.