Principal value

Consider the complex logarithm function . It is defined as the complex number such that :e^w = z. Now, for example, say we wish to find . 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 in the complex plane and in particular its argument \arg i. We can rotate counterclockwise \pi/2 radians from 1 to reach initially, but if we rotate further another 2\pi we reach again. So, we can conclude that i(\pi/2 + 2\pi) is also a solution for . It becomes clear that we can add any multiple of 2\pi 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 :\log{z} = \ln + i\left(\mathrm{arg}\ z \right) = \ln + i\left(\mathrm{Arg}\ z+2\pi k\right) for an integer , where is the (principal) argument of defined to lie in the interval (-\pi,\ \pi]. 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 \pi 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. ==General case==

In general, if is multiple-valued, the principal branch of is denoted :\mathrm{pv}\,f(z) such that for in the domain of , 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 + i\left(\mathrm{arg}\ z\right). Now, 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 (with the leading capital A). Using instead of , we obtain the principal value of the logarithm, and we write :\mathrm{pv}\log{z} = \mathrm{Log}\,z = \ln + i\left(\mathrm{Arg}\,z\right). Square root For a complex number z = r e^{i \phi}\, the principal value of the square root is: :\mathrm{pv}\sqrt{z} = \exp\left(\frac{\mathrm{pv}\log z}{2}\right) = \sqrt{r}\, e^{i \phi / 2} with argument -\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 \phi = \pi. Inverse trigonometric and inverse hyperbolic functions 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. Complex argument 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| function. The value of will be in the interval (-\pi, \pi]. In comparison, is typically in (\tfrac{-\pi}{2}, \tfrac{\pi}{2}]. ==See also==