Principal branch

In mathematics, a principal branch is a function which selects one 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

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 Template:Math.

For example, take the relation Template:Math, where Template:Math is any positive real number.

This relation can be satisfied by any value of Template:Math equal to a square root of Template:Math (either positive or negative). By convention, Template:Sqrt is used to denote the positive square root of Template:Math.

In this instance, the positive square root function is taken as the principal branch of the multi-valued relation Template:Math.

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 Template:Math 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 Template:Math is any integer and Template:Math continues the values of the Template:Math-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 Template:Math-function <math>(-\pi,\; \pi]</math>, covering all four quadrants in the complex plane.

Any number Template:Math defined by such criteria has the property that Template:Math.

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 Template:Math and Template:Math. These are the chosen principal values.

This is the principal branch of the log function. Often it is defined using a capital letter, Template:Math.

See also

• Branch point
• Branch cut
• Complex logarithm
• Riemann surface
• Square root#Principal square root of a complex number

External links

• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:PrincipalBranch%7CPrincipalBranch.html}} |title = Principal Branch |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Branches of Complex Functions Module by John H. Mathews