Principal branch

 ( Complex Analysis ) 

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

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Examples

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Trigonometric inverses 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

$\backslash arccos:-1,+1\backslash rightarrow0,\backslash pi$.

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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 .

For example, take the relation , where is any positive real number.

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

In this instance, the positive square root function is taken as the principal branch of the multi-valued relation .

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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 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 is any integer and continues the values of the -function from their principal value range $(-\backslash pi/2,\backslash ;\; \backslash pi/2]$, corresponding to $a\; >\; 0$ into the principal value range of the -function $(-\backslash pi,\backslash ;\; \backslash pi]$, covering all four quadrants in the complex plane.

Any number defined by such criteria has the property that .

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 and . These are the chosen .

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

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See also

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

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External links

• Branches of Complex Functions Module by John H. Mathews

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