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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
: see branch cut.

One way to view a principal branch is to look specifically at the exponential function
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
, and the logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
, as it is defined in complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
.

The exponential function is single-valued, where is defined as:

where .

However, the periodic nature of the trigonometric functions involved makes it clear that the logarithm is not so uniquely determined.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, 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
Complex plane

In mathematics, the complex plane is a geometric representation of the complex numbersestablished by the real axis and the orthogonal imaginary axis....
: see branch cut.

One way to view a principal branch is to look specifically at the exponential function
Exponential function

The exponential function is a function in mathematics. The application of this function to a value x is written as exp. Equivalently, this can be written in the form ex, where e is the mathematical constant that is the base of the natural logarithm and that is also known as Euler's number....
, and the logarithm
Logarithm

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
, as it is defined in complex analysis
Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics investigating Function of complex numbers....
.

The exponential function is single-valued, where is defined as:

where .

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:

and

where k is any integer.

Any number log(z) 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 −p and p. These are the chosen principal value
Principal value

In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is Single-valued function....
s.

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 = x1/2, where x is any positive real number.

This relation can be satisfied by any value of y equal to a square root
Square root

In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x....
 of x (either positive or negative). When y is taken to be the positive square root, we write .

In this instance, the positive square root function is taken as the principal branch of the multi-valued relation x1/2.

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

Trigonometry is a branch of mathematics that deals with triangle s, particularly those plane triangles in which one angle has 90 degrees . Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships....
 functions.

See also

  • Branch point
    Branch point

    In the mathematics field of complex analysis, a branch point of a multivalued function is a point such that the function is discontinuous when going around an arbitrarily small circuit around this point ....
  • Complex logarithm
    Complex logarithm

    In complex analysis, a complex logarithm function is an "inverse function" of the complex exponential function, just as the natural logarithm ln x is the inverse of the exponential function ex....
  • Riemann surface
    Riemann surface

    In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold....


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