In
mathematics, the dilogarithm (or Spence's function), denoted as Li2(z), is a particular case of the
polylogarithm. Two related
special functions are referred to as Spence's function, the dilogarithm itself:
and its reflection.
For |z| < 1, an
infinite series also applies (the integral definition constitutes its analytical extension to the
complex plane):
Alternatively, the dilogarithm function is sometimes defined as
William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century.[2] He was at school with
John Galt,[3] 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 .
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:
In
mathematics, the dilogarithm (or Spence's function), denoted as Li2(z), is a particular case of the
polylogarithm. Two related
special functions are referred to as Spence's function, the dilogarithm itself:
and its reflection.
For |z| < 1, an
infinite series also applies (the integral definition constitutes its analytical extension to the
complex plane):
Alternatively, the dilogarithm function is sometimes defined as
William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century.[2] He was at school with
John Galt,[3] 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 .
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: