Principal Value - Motivation

Motivation

Consider the complex logarithm function log z. It is defined as the complex number w such that

Now, for example, say we wish to find log i. This means we want to solve

for w. Clearly iπ/2 is a solution. But is it the only solution?

Of course, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument arg i. We can rotate counterclockwise π/2 radians from 1 to reach i initially, but if we rotate further another 2π we reach i again. So, we can conclude that i(π/2 + 2π) is also a solution for log i. It becomes clear that we can add any multiple of 2πi to our initial solution to obtain all values for log i.

But this has a consequence that may be surprising in comparison of real valued functions: log i does not have one definite value! For log z, we have

for an integer k, where Arg z is the (principal) argument of z defined to lie in the interval . Each value of k determines what is known a branch (or sheet), a single-valued component of the multiple-valued log function.

The branch corresponding to k=0 is known as the principal branch, and along this branch, the values the function takes are known as the principal values.