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 Template:Vanchores 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 Template:Tmath. The principal square root of a non real complex number is the one with an argument in the interval Template:Tmath, and the principal square root of a negative real number Template:Tmath is Template:Tmath.

Motivation

Consider the complex logarithm function Template:Math. It is defined as the complex number Template:Mvar such that

$e^w=z.$

Now, for example, say we wish to find Template:Math. This means we want to solve

$e^w=i$

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

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

But this has a consequence that may be surprising in comparison of real valued functions: Template:Math does not have one definite value. For Template:Math, we have

$\log z=\ln |z|+i\left(\mathrm{a}\mathrm{r}\mathrm{g}z\right)=\ln |z|+i(\mathrm{A}\mathrm{r}\mathrm{g}z+2\pi k)$

for an integer Template:Mvar, where Template:Math is the (principal) argument of Template:Mvar defined to lie in the interval $(-\pi ,\pi ]$. Each value of Template:Mvar 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 $\pi$ as a possible value for Template:Math. With this branch cut, the single-branch function is continuous and analytic everywhere in its domain.

The branch corresponding to Template:Math is known as the principal branch, and along this branch, the values the function takes are known as the principal values.

General case

In general, if Template:Math is multiple-valued, the principal branch of Template:Mvar is denoted

$\mathrm{p}\mathrm{v}f(z)$

such that for Template:Mvar in the domain of Template:Mvar, Template:Math is single-valued.

Principal values of standard functions

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.

Logarithm function

We have examined the logarithm function above, i.e.,

$\log z=\ln |z|+i\left(\mathrm{a}\mathrm{r}\mathrm{g}z\right).$

Now, Template:Math is intrinsically multivalued. One often defines the argument of some complex number to be between $-\pi$ (exclusive) and $\pi$ (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch Template:Math (with the leading capital A). Using Template:Math instead of Template:Math, we obtain the principal value of the logarithm, and we write

$\mathrm{p}\mathrm{v}\log z=\mathrm{L}\mathrm{o}\mathrm{g}z=\ln |z|+i\left(\mathrm{A}\mathrm{r}\mathrm{g}z\right).$

Square root

For a complex number $z=r{e}^{i\varphi }$ the principal value of the square root is:

$\mathrm{p}\mathrm{v}\sqrt{z}=\exp \left(\frac{\mathrm{p}\mathrm{v}\log z}{2}\right)=\sqrt{r}{e}^{i\varphi /2}$

with argument $-\pi <\varphi \le \pi .$ 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 $\varphi =\pi .$

Inverse trigonometric and inverse hyperbolic functions

Inverse trigonometric functions (Template:Math, Template:Math, Template:Math, etc.) and inverse hyperbolic functions (Template:Math, Template:Math, Template:Math, etc.) can be defined in terms of logarithms and their principal values can be defined in terms of the principal values of the logarithm.

Complex argument

comparison of atan and atan2 functions

The principal value of complex number argument measured in radians can be defined as:

• values in the range $[0,2\pi )$
• values in the range $(-\pi ,\pi ]$

For example, many computing systems include an [[atan2|Template:Math]] function. The value of Template:Math will be in the interval $(-\pi ,\pi ].$ In comparison, Template:Math is typically in $(\frac{-\pi }{2},\frac{\pi }{2}].$

See also

• Principal branch
• Branch point