Principal value

In complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive real number. For example, 4 has two square roots: 2 and –2; of these the positive root, 2, is considered the principal root and is denoted as ${\displaystyle {\sqrt {4}}.}$

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 ${\displaystyle (-\pi ,\ \pi ]}$. Each value of k determines what is known as a branch (or sheet), a single-valued component of the multiple-valued log function.

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