Principal branch

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 $z\; =\; a\; +\; bi$ .

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:

:$Re(log\; z)=log\; sqrt\{a^2\; +\; b^2\}$

and

:$Im(log\; z)\; =\; arctan(b/a)\; +\; 2pi\; k$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.

ee also

*Branch point
*Riemann surface
*Complex analysis

External links

*
* [http://math.fullerton.edu/mathews/c2003/ComplexFunBranchMod.html Branches of Complex Functions Module by John H. Mathews]