Principal value
In mathematics, specifically complex analysis, a multivalued function has often the property that, near almost every point, the graph of the function is the disjoint union of one or several graphs of smooth functions, which are called branches of the multivalued functions. In the case of complex analytic functions, these branches can be prolongated to smooth functions that are defined in the whole complex plane except a finite number of points, and are equal to one value of the multivalued function in their domain. Often, a branch refers value specifically to such a maximal branch.
The principal branch of a multivariate function is one of these maximal branches that is selected once for all. Typically, the principal branch is the branch that takes a real value for small positive values of the variable.
A principal value is the value at a point of the function defined by the principal branch. In many cases, the principal value at a point of a multivalued function is distinguished from the other values by being the one whose argument has the smallest absolute value, and, when there are two such values, the one with positive real part.
A simple example is given by the square root function: every nonzero complex number has two square roots. The principal value of the square root of a positive real number is the positive square root denoted . The principal square root of a non real complex number is the one with an argument in the interval , and the principal square root of a negative real number is .
