Principal branch

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In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut.

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 exp(z) is defined as:

exp(z) = exp(a) cos(b) + i exp(a) sin(b), where z = a + bi .

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:

Re(log(z)) = log(√(a² + b²))

and

Im(log(z)) = arctan(b/a) + 2πk, where k is any integer.

Any number log(z) defined by such criteria has the property that exp(log(z))=z.

In this manner log function is a multi-valued function. A branch cut, usually along the negative real axis, can limit the imaginary part so it lies between −π and π.

This is the principal branch of the log function. Often it is defined using a capital letter, Log(z).

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). When y is taken to be the positive square root, we write y = √x.

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

Principal branches are also used in the definition of many inverse trigonometric functions.