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 4 . {displaystyle {sqrt {4}}.} 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 4 . {displaystyle {sqrt {4}}.} Consider the complex logarithm function log z. It is defined as the complex number w such that