Principal value

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.

Consider the complex logarithm function log z.

However, there are other solutions, which is evidenced by considering the position of i in the complex plane and in particular its argument

to our initial solution to obtain all values for log i.

For log z, we have for an integer k, where Arg z is the (principal) argument of z defined to lie in the interval

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

With this branch cut, the single-branch function is continuous and analytic everywhere in its domain.

In general, if f(z) is multiple-valued, the principal branch of f is denoted such that for z in the domain of f, pv f(z) is single-valued.

Complex valued elementary functions can be multiple-valued over some domains.

We have examined the logarithm function above, i.e., Now, arg z is intrinsically multivalued.

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 (arcsin, arccos, arctan, etc.)

and inverse hyperbolic functions (arsinh, arcosh, artanh, 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 atan2(y, x) function.