Editing Principal value (section)

==General case==
In general, if {{math|''f''(''z'')}} is multiple-valued, the principal branch of {{mvar|f}} is denoted
:<math>\mathrm{pv}\,f(z)</math>
such that for {{mvar|z}} in the [[Domain of a function|domain]] of {{mvar|f}}, {{math|pv ''f''(''z'')}} is single-valued.

===Principal values of standard functions===
Complex valued [[List of mathematical functions|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., 
:<math>\log{z} = \ln{|z|} + i\left(\mathrm{arg}\ z\right).</math>
Now, {{math|arg ''z''}} is intrinsically multivalued. One often defines the argument of some complex number to be between <math>-\pi</math> (exclusive) and <math>\pi</math> (inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch {{math|Arg ''z''}} (with the leading capital A). Using {{math|Arg ''z''}} instead of {{math|arg ''z''}}, we obtain the principal value of the logarithm, and we write<ref>{{Cite book |last1=Zill |first1=Dennis |url=https://books.google.com/books?id=YKZqY8PCNo0C |title=A First Course in Complex Analysis with Applications |last2=Shanahan |first2=Patrick |date=2009 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-5772-4 |pages=166 |language=en}}</ref>
:<math>\mathrm{pv}\log{z} = \mathrm{Log}\,z = \ln{|z|} + i\left(\mathrm{Arg}\,z\right).</math>

====Square root====
For a complex number <math>z = r e^{i \phi}\,</math> the principal value of the [[square root]] is:

:<math>\mathrm{pv}\sqrt{z} = \exp\left(\frac{\mathrm{pv}\log z}{2}\right) = \sqrt{r}\, e^{i \phi / 2}</math>

with [[Arg (mathematics)|argument]] <math>-\pi < \phi \le \pi.</math>  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 <math>\phi = \pi.</math>

====Inverse trigonometric and inverse hyperbolic functions====
Inverse trigonometric functions ({{math|arcsin}}, {{math|arccos}}, {{math|arctan}}, etc.) and inverse hyperbolic functions ({{math|arsinh}}, {{math|arcosh}}, {{math|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.

====Complex argument====
[[File:Atan2atan.png|right|thumb|comparison of [[Inverse trigonometric functions|atan]] and [[atan2]] functions]]

The principal value of [[Arg (mathematics)|complex number argument]] measured in [[radian]]s can be defined as:
* values in the range <math>[0, 2\pi)</math>
* values in the range <math>(-\pi, \pi]</math>

For example, many computing systems include an [[atan2|{{math|atan2(''y'', ''x'')}}]] function.  The value of {{math|atan2(imaginary_part(''z''), real_part(''z''))}} will be in the interval <math>(-\pi, \pi].</math>  In comparison, {{math|atan ''y''/''x''}} is typically in <math>(\tfrac{-\pi}{2}, \tfrac{\pi}{2}].</math>