Trigonometric inverses

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

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

or that

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

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.

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:

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

where ${\displaystyle z=a+ib}$.

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:

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

and

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

where k is any integer and atan2 continues the values of the arctan(b/a)-function from their principal value range ${\displaystyle (-\pi /2,\;\pi /2]}$, corresponding to ${\displaystyle a>0}$ into the principal value range of the arg(z)-function ${\displaystyle (-\pi ,\;\pi ]}$, 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.