section_path: Non-integer powers of complex numbers/Complex exponentiation/Principal value
section_name: Principal value
paragraph_text: Principal value
  
  The principal value of the complex logarithm is the unique continuous function, commonly denoted log, {\displaystyle \log,} such that, for every nonzero complex number z,
  
  and the argument of z satisfies
  
  The principal value of the complex logarithm is not defined for z = 0, {\displaystyle z=0,} it is discontinuous at negative real values of z, and it is holomorphic (that is, complex differentiable) elsewhere. If z is real and positive, the principal value of the complex logarithm is the natural logarithm: log z = ln z. {\displaystyle \log z=\ln z.}
  
  The principal value of z w {\displaystyle z^{w}} is defined as z w = e w log z, {\displaystyle z^{w}=e^{w\log z},} where log z {\displaystyle \log z} is the principal value of the logarithm.
  
  The function (z, w) → z w {\displaystyle (z,w)\to z^{w}} is holomorphic except in the neighbourhood of the points where z is real and nonpositive.
  
  If z is real and positive, the principal value of z w {\displaystyle z^{w}} equals its usual value defined above. If w = 1 / n, {\displaystyle w=1/n,} where n is an integer, this principal value is the same as the one defined above.