Logarithm of the imaginary unit
The logarithm of the imaginary unit i is the multi-valued function
Here the natural logarithm is written as log.
Explanation
A complex number w is called a logarithm of z, so w = log z, if ew = z.
We speak of a logarithm because for z there are infinitely many numbers w that act as logarithms. They differ from each other by an integer multiple of 2πi. This is because e2kπ i = 1. If we write z as
z = r · eiφ
with absolute value r and argument φ, then each of the numbers
is a logarithm of z. It is common to define the argument φ in such a way that –π < φ ≤ π. The value of the logarithm for k = 0, is called the principle value of the logarithm.
The imaginary unity is a complex number. Because z = i lies on the unit circle with radius r = 1 and a quarter circle turned φ = ½π. The logarithm of i therefore has a principal value of
log (i) = log (1) + ½πi = ½πi