Introduction to the Mathematical and Statistical Foundations of Econometrics
The Complex Logarithm
Like the natural logarithm ln( ), the complex logarithm log(z), z є C is a complex number a + i ■ b = log(z) such that exp(a + i ■ b) = z; hence, it follows from (Ш.15) that z = exp(a)[cos(b) + i ■ sin(b)] and consequently, because
| exp(-a) ■ z| = | cos(b) + i ■ sin(b)| = у cos2(b) + sin2(b) = 1,
we have that exp(a) = |z| andexp(i ■ b) = z/|z|. The first equation has a unique solution, a = ln(|z|), as long as z = 0. The second equation reads as
cos(b) + i ■ sin(b) = (Re(z) + i ■ Im(z))/|z|; (III.16)
hence cos(b) = Re(z)/|z|, sin(b) = Im(z)/|z|, and thus b = arctan(Im(z)/ Re(z)). However, equation (III.16) also holds if we add or subtract multiples of n to or from b because tan(b) = tan(b + m ■ n) for arbitrary integers m; hence,
log(z) = ln(|z|) + i ■ [arctan(Im(z)/Re(z)) + mn],
m = 0, ±1, ±2, ±3,.... (III.17)
Therefore, the complex logarithm is not uniquely defined.
The imaginary part of (III.17) is usually denoted by
arg(z) = arctan(Im(z)/Re(z)) + mn, m = 0, ±1, ±2, ±3,________
It is the angle in radians indicated in Figure III.1 eventually rotated multiples of 180° clockwise or counterclockwise: Note that Im(z)/Re(z) is the tangents of the angle arg(z); hence, arctan (Im(z)/Re(z)) is the angle itself.
With the complex exponential function and logarithm defined, we can now define the power zw as the complex number a + i ■ b such that a + i ■ b = exp(w ■ log(z)), which exists if |z| > 0. Consequently, DeMoivre’s formula carries over to all real-valued powers n:
[cos(b) + i ■ sin(b)]n = (elb)n = e1 n b = cos(n ■ b) + i ■ sin(n ■ b).
