principal branch

File:Principle branch of arg on Riemann.svg

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

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

or that

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

A more familiar principal branch function, limited to real numbers, is that of a positive real number raised to the power of {{math|1/2}}.

For example, take the relation {{math|y {{=}} x^1/2}}, where {{math|x}} is any positive real number.

This relation can be satisfied by any value of {{math|y}} equal to a square root of {{math|x}} (either positive or negative). By convention, {{sqrt|x}} is used to denote the positive square root of {{math|x}}.

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

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 {{math|e^z}} is defined as:

:$e^z\; =\; e^a\; \backslash cos\; b\; +\; i\; e^a\; \backslash sin\; b$

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:

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

and

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

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

Any number {{math|log z}} defined by such criteria has the property that {{math|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 {{math|−π}} and {{math|π}}. These are the chosen principal values.

This is the principal branch of the log function. Often it is defined using a capital letter, {{math|Log z}}.

• Branch point
• Branch cut
• Complex logarithm
• Riemann surface
• Square root#Principal square root of a complex number

• {{MathWorld | urlname= PrincipalBranch | title= Principal Branch }}
• [https://web.archive.org/web/20061209014913/http://math.fullerton.edu/mathews/c2003/ComplexFunBranchMod.html Branches of Complex Functions Module by John H. Mathews]

Category:Complex analysis