• Source: Principal branch

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






= Trigonometric inverses

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




arcsin
:
[
−
1
,
+
1
]
→

[

−


π
2


,


π
2



]



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


or that




arccos
:
[
−
1
,
+
1
]
→
[
0
,
π
]


{\displaystyle \arccos :[-1,+1]\rightarrow [0,\pi ]}

.




= 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, √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

=
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


{\displaystyle e^{z}=e^{a}\cos b+ie^{a}\sin b}


where



z
=
a
+
i
b


{\displaystyle z=a+ib}

.
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
⁡



a

2


+

b

2






{\displaystyle \operatorname {Re} (\log z)=\log {\sqrt {a^{2}+b^{2}}}}


and




Im
⁡
(
log
⁡
z
)
=
atan2
⁡
(
b
,
a
)
+
2
π
k


{\displaystyle \operatorname {Im} (\log z)=\operatorname {atan2} (b,a)+2\pi k}


where k is any integer and atan2 continues the values of the arctan(b/a)-function from their principal value range



(
−
π

/

2
,

π

/

2
]


{\displaystyle (-\pi /2,\;\pi /2]}

, corresponding to



a
>
0


{\displaystyle a>0}

into the principal value range of the arg(z)-function



(
−
π
,

π
]


{\displaystyle (-\pi ,\;\pi ]}

, covering all four quadrants in the complex plane.
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.




See also


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




External links


Weisstein, Eric W. "Principal Branch". MathWorld.
Branches of Complex Functions Module by John H. Mathews

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