Logarithm of a complex number
The logarithm of a complex number is a multi-valued function
ln z = ln r + ( φ+ 2kπ) i
Explanation
You write a complex number with polar coordinates as z = r · eiφ. If we assume that the logarithm of this produces the complex number x + i y is, then you get
where
so that
ln z = ln r + (φ + 2kπ) i
The logarithm of a complex number has an infinite number of values, that all have the same real part ln r and the imaginary part differs a multiple of 2π from each other.
For k = 0 you get the principle branch.
Example 1
Negative numbers are a special case of complex numbers. Thus z = −1 is a complex number on the unit circle with radius r = 1 and a semicircle rotated φ = π. The logarithm of −1 has a principal value of
ln (−1) = ln (1) + πi = πi