Principal branch

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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[edit]

Trigonometric inverses[edit]

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[edit]

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, 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 x1/2.

Complex logarithms[edit]

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 ez is defined as:

e^z = e^a \cos b + i e^a \sin b

where z = a + i b .

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 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.

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