Principal Value

Get Principal Value essential facts below. View Videos or join the Principal Value discussion. Add Principal Value to your PopFlock.com topic list for future reference or share this resource on social media.

## Motivation

## General case

### Principal values of standard functions

#### Logarithm function

#### Square root

#### Complex argument

## See also

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Principal Value

This article does not cite any sources. (December 2009) (Learn how and when to remove this template message) |

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. 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

Consider the complex logarithm function log *z*. It is defined as the complex number *w* such that

Now, for example, say we wish to find log *i*. This means we want to solve

for *w*. Clearly *i*?/2 is a solution. But is it the only solution?

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

But this has a consequence that may be surprising in comparison of real valued functions: log *i* does not have one definite value! For log *z*, we have

for an integer *k*, where Arg *z* is the (principal) argument of *z* defined to lie in the interval . As the principal argument is unique for a given complex number *z*, is not included 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.

The branch corresponding to *k* = 0 is known as the *principal branch*, and along this branch, the values the function takes are known as the *principal values*.

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. 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, arg *z* 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 Arg *z* (with the leading capital A). Using Arg *z* instead of arg *z*, 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

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

- values in the range
- values in the range

To compute these values one can use functions :

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Popular Products

Music Scenes

Popular Artists