In mathematics, specifically 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. A simple 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 ${\displaystyle {\sqrt {4}}.}$

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

${\displaystyle e^{w}=z.}$

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

${\displaystyle e^{w}=i}$

for ${\displaystyle w}$. The value ${\displaystyle i\pi /2}$ is a solution.

However, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument ${\displaystyle \arg i}$. We can rotate counterclockwise ${\displaystyle \pi /2}$ radians from 1 to reach i initially, but if we rotate further another ${\displaystyle 2\pi }$ we reach i again. So, we can conclude that ${\displaystyle i(\pi /2+2\pi )}$ is also a solution for log i. It becomes clear that we can add any multiple of ${\displaystyle 2\pi }$ 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

${\displaystyle \log {z}=\ln {|z|}+i\left(\mathrm {arg} \ z\right)=\ln {|z|}+i\left(\mathrm {Arg} \ z+2\pi k\right)}$

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. When the focus is on a single branch, sometimes a branch cut is used; in this case removing the non-positive real numbers from the domain of the function and eliminating ${\displaystyle \pi }$ as a possible value for Arg z. With this branch cut, the single-branch function is continuous and analytic everywhere in its domain.

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.

In general, if f(z) is multiple-valued, the principal branch of f is denoted

${\displaystyle \mathrm {pv} \,f(z)}$

such that for z in the domain of f, pv f(z) is single-valued.

• Principal branch
• Branch point