In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, the principal values of a
multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to a ...
are the values along one chosen
branch
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk (botany), trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' ...
of that
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
, so that it is
single-valued. The simplest case arises in taking the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
of a positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. For example, 4 has two square roots: 2 and −2; of these the positive root, 2, is considered the principal root and is denoted as
Motivation
Consider the
complex logarithm function log ''z''. It is defined as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
''w'' such that
:
Now, for example, say we wish to find log ''i''. This means we want to solve
:
for ''w''. Clearly ''i''π/2 is a solution. But is it the only solution?
Of course, there are other solutions, which is evidenced by considering the position of ''i'' in the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
and in particular its
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
arg ''i''. We can rotate counterclockwise π/2 radians from 1 to reach ''i'' initially, but if we rotate further another 2π we reach ''i'' again. So, we can conclude that ''i''(π/2 + 2π) is ''also'' a solution for log ''i''. It becomes clear that we can add any multiple of 2π''i'' to our initial solution to obtain all values for log ''i''.
But this has a consequence that may be surprising in comparison of real valued functions: log ''i'' does not have one definite value. For log ''z'', we have
:
for an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
''k'', where Arg ''z'' is the (principal) argument of ''z'' defined to lie in the
interval . As the principal argument is unique for a given complex number ''z'',
is not included in the interval. Each value of ''k'' determines what is known as a ''
branch
A branch, sometimes called a ramus in botany, is a woody structural member connected to the central trunk (botany), trunk of a tree (or sometimes a shrub). Large branches are known as boughs and small branches are known as twigs. The term '' ...
'' (or ''sheet''), a single-valued component of the multiple-valued log function.
The branch corresponding to ''k'' = 0 is known as the ''principal branch'', and along this branch, the values the function takes are known as the ''principal values''.
General case
In general, if ''f''(''z'') is multiple-valued, the principal branch of ''f'' is denoted
:
such that for ''z'' in the
Domain of a function, domain of ''f'', pv ''f''(''z'') is single-valued.
Principal values of standard functions
Complex valued
elementary functions
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and ...
can be multiple-valued over some domains. The principal value of some of these functions can be obtained by decomposing the function into simpler ones whereby the principal value of the simple functions are straightforward to obtain.
Logarithm function
We have examined the
logarithm function
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
above, i.e.,
:
Now, arg ''z'' is intrinsically multivalued. One often defines the argument of some complex number to be between
(exclusive) and
(inclusive), so we take this to be the principal value of the argument, and we write the argument function on this branch Arg ''z'' (with the leading capital A). Using Arg ''z'' instead of arg ''z'', we obtain the principal value of the logarithm, and we write
:
Square root
For a complex number
the principal value of the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
E ...
is:
:
with
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
Complex argument
![Atan2atan](https://upload.wikimedia.org/wikipedia/commons/c/c7/Atan2atan.png)
The principal value of
complex number argument measured in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s can be defined as:
* values in the range