Principal branches are used in the definition of many inverse trigonometric functions, such as the selection either to define that cos-1:ℝ↦(-π,π] or that cos-1:ℝ↦[0,2π).
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 = x1/2, 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, √ 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 x1/2.
The exponential function is single-valued, where ez is defined as:
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:
where k is any integer and atan2 is arctangent with the appropriate sign correction.
Any number log z defined by such criteria has the property that elog 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.