Total least squares: Difference between revisions

Revision as of 23:03, 27 January 2014

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 $e^{z}$ is defined as:

e^{z} = e^{a} \cos b + i e^{a} \sin b

where $z = a + b i$ .

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:

ℜ (\log z) = \log \sqrt{a^{2} + b^{2}}

and

ℑ (\log z) = \arctan (b / a) + 2 π k

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 values.

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 $y = \sqrt{x}$ .

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 trigonometric functions.

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Branches of Complex Functions Module by John H. Mathews Template:Dead link

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+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''<sup>log(''z'')</sup>&nbsp;=&nbsp;''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 &minus;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''<sup>1/2</sup>, 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''<sup>1/2</sup>.
+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]]

Total least squares: Difference between revisions

Revision as of 23:03, 27 January 2014

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