Logarithm of a negative number
The logarithm of a negative number is the multi-valued function
Here the natural logarithm is written as log.
Explanation
Every negative number is a complex number. With polar coordinates (r, θ) you can describe negative numbers, because they lie in the complex plane on the real x-axis, where θ = 0, ±π, ±2π, ... is.
The definition of the logarithm can be extended to negative and complex arguments. A logarithm is the inverse of an exponential function. For the complex number w applies
ew = z with as inverse w = log 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
with absolute value r and argument φ, then each of the numbers
is a logarithm of z. The logarithm for complex numbers z is a multi-valued function
The value of the logarithm for n = 0, is called the principle value of the logarithm. The argument of z in the interval [0, 2π) is called the principle value. The interval (−π, π] is also chosen as the principle value.
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 φp = π. The logarithm of −1 therefore has a principal value of