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 ...
, Spence's function, or dilogarithm, denoted as , is a particular case of the
polylogarithm
In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function of order and argument . Only for special values of does the polylogarithm reduce to an elementary function such as the natur ...
. Two related
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
are referred to as Spence's function, the dilogarithm itself:
:
and its reflection.
For , an
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
also applies (the integral definition constitutes its analytical extension to 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 ...
):
:
Alternatively, the dilogarithm function is sometimes defined as
:
In
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P'' ...
the dilogarithm can be used to compute the volume of an
ideal simplex
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
. Specifically, a simplex whose vertices have
cross ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
has
hyperbolic volume
In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv ...
:
The function is sometimes called the Bloch-Winger function.
Lobachevsky's function and
Clausen's function
In mathematics, the Clausen function, introduced by , is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other forms. It is intimate ...
are closely related functions.
William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with
John Galt
John Galt () is a character in Ayn Rand's novel ''Atlas Shrugged'' (1957). Although he is not identified by name until the last third of the novel, he is the object of its often-repeated question "Who is John Galt?" and of the quest to discover ...
,
who later wrote a biographical essay on Spence.
Analytic structure
Using the former definition above, the dilogarithm function is analytic everywhere on the complex plane except at
, where it has a logarithmic branch point. The standard choice of branch cut is along the positive real axis
. However, the function is continuous at the branch point and takes on the value
.
Identities
:
[Zagier]
:
:
:
:
Particular value identities
:
:
:
:
:
:
Special values
:
:
:
:
where
is the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
:
:
:
:
:
In particle physics
Spence's Function is commonly encountered in particle physics while calculating radiative corrections. In this context, the function is often defined with an absolute value inside the logarithm:
:
Notes
References
*
*
*
*
*
*
Further reading
*
External links
NIST Digital Library of Mathematical Functions: Dilogarithm* {{MathWorld, title=Dilogarithm, urlname=Dilogarithm
Special functions