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 âÂÂ2; of these the positive root, 2, is considered the principal root and is denoted as
Consider the complex logarithm function . It is defined as the complex number such that
Now, for example, say we wish to find . This means we want to solve
for . The value is a solution.
However, there are other solutions, which is evidenced by considering the position of in the complex plane and in particular its argument . We can rotate counterclockwise radians from 1 to reach initially, but if we rotate further another we reach again. So, we can conclude that is also a solution for . It becomes clear that we can add any multiple of to our initial solution to obtain all values for .
But this has a consequence that may be surprising in comparison of real valued functions: does not have one definite value. For , we have
for an integer , where is the (principal) argument of defined to lie in the interval . Each value of 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 as a possible value for . With this branch cut, the single-branch function is continuous and analytic everywhere in its domain.
The branch corresponding to is known as the principal branch, and along this branch, the values the function takes are known as the principal values.
In general, if is multiple-valued, the principal branch of is denoted
such that for in the domain of , is single-valued.
Complex valued elementary functions can be multiple-valued over some domains. The principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.
We have examined the logarithm function above, i.e.,
Now, is intrinsically multivalued. One often defines the argument of some complex number to be between (exclusive) and (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch (with the leading capital A). Using instead of , we obtain the principal value of the logarithm, and we write
For a complex number the principal value of the square root is:
with argument Sometimes a branch cut is introduced so that negative real numbers are not in the domain of the square root function and eliminating the possibility that
Inverse trigonometric functions (, , , etc.) and inverse hyperbolic functions (, , , etc.) can be defined in terms of logarithms and their principal values can be defined in terms of the principal values of the logarithm.
The principal value of complex number argument measured in radians can be defined as:
For example, many computing systems include an function. The value of will be in the interval In comparison, is typically in