Total least squares

2014-01-27T22:03:30Z

128.32.192.19: /* Algebraic point of view */

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

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^z</math> is defined as:

:<math>e^z=e^a \cos b +i e^a \sin b</math>
where <math>z = a + bi</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) = \arctan(b/a) + 2\pi k</math>
where ''k'' is any integer.

Any number log(''z'') defined by such criteria has the property that ''e''log(''z'') = ''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 −π and 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, Log(''z'').

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

For example, take the relation ''y'' = ''x''1/2, where ''x'' is any positive real number.

This relation can be satisfied by any value of y equal to a [[square root]] of ''x'' (either positive or negative). When y is taken to be the positive square root, we write <math>y = \sqrt x</math>.

In this instance, the positive square root function is taken as the principal branch of the multi-valued relation ''x''1/2.

Principal branches are also used in the definition of many inverse [[trigonometry|trigonometric]] functions.

==See also==
*[[Branch point]]
*[[Complex logarithm]]
*[[Riemann surface]]

==External links==
* {{MathWorld | urlname= PrincipalBranch | title= Principal Branch }}
* [http://math.fullerton.edu/mathews/c2003/ComplexFunBranchMod.html Branches of Complex Functions Module by John H. Mathews] {{Dead link|date=May 2011}}

[[Category:Complex analysis]]

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Total least squares