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 −2; of these the positive root, 2, is considered the principal root and is denoted as
\sqrt{4}.
Consider the complex logarithm function . It is defined as the complex number such that
ew=z.
ew=i
w
i\pi/2
\argi
\pi/2
2\pi
i(\pi/2+2\pi)
2\pi
But this has a consequence that may be surprising in comparison of real valued functions: does not have one definite value. For, we have
log{z}=ln{|z|}+i\left(arg z\right) =ln{|z|}+i\left(Arg z+2\pik\right)
(-\pi, \pi]
\pi
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
pvf(z)
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.,
log{z}=ln{|z|}+i\left(arg z\right).
-\pi
\pi
pvlog{z}=Logz=ln{|z|}+i\left(Argz\right).
For a complex number
z=rei
pv\sqrt{z}=\exp\left(
pvlogz | |
2 |
\right)=\sqrt{r}ei
-\pi<\phi\le\pi.
\phi=\pi.
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:
[0,2\pi)
(-\pi,\pi]
For example, many computing systems include an function. The value of will be in the interval
(-\pi,\pi].
(\tfrac{-\pi}{2},\tfrac{\pi}{2}].