Editing Principal branch

In [[mathematics]], a '''principal branch''' is a function which selects one [[branch point|branch]] ("slice") of a [[multi-valued function]].  Most often, this applies to functions defined on the [[complex plane]].

== Examples ==
[[File:Principle branch of arg on Riemann.svg|thumbnail|right|Principal branch of arg(z)]]

=== Trigonometric inverses ===

Principal branches are used in the definition of many [[inverse trigonometric functions]], such as the selection either to define that
:<math>\arcsin:[-1,+1]\rightarrow\left[-\frac{\pi}{2},\frac{\pi}{2}\right]</math>
or that
:<math>\arccos:[-1,+1]\rightarrow[0,\pi]</math>.

=== Exponentiation to fractional powers ===

A more familiar principal branch function, limited to real numbers, is that of a positive real number raised to the power of {{math|1/2}}.

For example, take the relation {{math|''y'' {{=}} ''x''<sup>1/2</sup>}}, where {{math|''x''}} is any positive real number.

This relation can be satisfied by any value of {{math|''y''}} equal to a [[square root]] of {{math|''x''}} (either positive or negative).  By convention, {{sqrt|x}} is used to denote the positive square root of {{math|''x''}}.

In this instance, the positive square root function is taken as the principal branch of the multi-valued relation {{math|''x''<sup>1/2</sup>}}.

=== Complex logarithms ===

One way to view a principal branch is to look specifically at the [[exponential function]], and the [[logarithm]], as it is defined in [[complex analysis]].

The exponential function is single-valued, where {{math|''e<sup>z</sup>''}} is defined as:

:<math>e^z = e^a \cos b + i e^a \sin b</math>
where <math>z = a + i b</math>.

However, the periodic nature of the trigonometric functions involved makes it clear that the logarithm is not so uniquely determined.  One way to see this is to look at the following:

:<math>\operatorname{Re} (\log z) = \log \sqrt{a^2 + b^2}</math>

and

:<math>\operatorname{Im} (\log z) = \operatorname{atan2}(b, a) + 2 \pi k</math>
where {{math|''k''}} is any integer and {{math|[[atan2]]}} continues the values of the {{math|arctan(b/a)}}-function from their principal value range <math>(-\pi/2,\; \pi/2]</math>, corresponding to <math>a > 0</math> into the principal value range of the {{math|arg(z)}}-function <math>(-\pi,\; \pi]</math>, covering all four quadrants in the complex plane.

Any number {{math|log ''z''}} defined by such criteria has the property that {{math|''e''<sup>log ''z''</sup> {{=}} ''z''}}.

In this manner log function is a [[multi-valued function]] (often referred to as a "multifunction" in the context of complex analysis).  A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between {{math|−π}} and {{math|π}}. These are the chosen [[principal value]]s.

This is the principal branch of the log function.  Often it is defined using a capital letter, {{math|Log ''z''}}.

==See also==
*[[Branch point]]
* [[Branch cut]]
*[[Complex logarithm]]
*[[Riemann surface]]
*[[Square root#Principal square root of a complex number]]

==External links==
* {{MathWorld | urlname= PrincipalBranch | title= Principal Branch }}
* [https://web.archive.org/web/20061209014913/http://math.fullerton.edu/mathews/c2003/ComplexFunBranchMod.html Branches of Complex Functions Module by John H. Mathews]

[[Category:Complex analysis]]