Principle value (of a logarithm)
The logarithm of a complex number z in the interval [0, 2π) is called the principle value. The interval (−π, π] can also be taken.
Explanation
The logarithm of a complex number z = r · eiφ is
ln z = ln r + (φ + 2kπ) i
You write a complex number in polar form as z = r · eiφ. The logarithm of this number can be expressed as the complex number x + i y, so
Where
so that
ln z = ln r + (φ + 2kπ) i
The logarithm of a complex number has infinitely many values, that all have the same real part ln r and that differ a multiple of 2π in the imaginary part.
For k = 0 you get the principle value.
Example 1
The logarithm of −1 has a principle value of