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.

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Examples

Principal branch of arg(z)

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Trigonometric inverses

Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that

- $\arcsin :[-1,+1]\rightarrow \left[-{\frac {\pi }{2)),{\frac {\pi }{2))\right]$

or that

- $\arccos :[-1,+1]\rightarrow [0,\pi ]$.

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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 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). By convention, √x is used to denote the positive square root of *x*.

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

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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 *e*^{z} is defined as:

- $e^{z}=e^{a}\cos b+ie^{a}\sin b$

where $z=a+ib$.

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:

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

and

- $\operatorname {Im} (\log z)=\operatorname {atan2} (b,a)+2\pi k$

where *k* is any integer and atan2 continues the values of the arctan(b/a)-function from their principal value range $(-\pi /2,\;\pi /2]$, corresponding to $a>0$ into the principal value range of the arg(z)-function $(-\pi ,\;\pi ]$, covering all four quadrants in the complex plane.

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*.