Spence's function

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The dilogarithm along the real axis

In mathematics, Spence's function, or dilogarithm, 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 an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane):

Alternatively, the dilogarithm function is sometimes defined as

In hyperbolic geometry the dilogarithm occurs as the hyperbolic volume of an ideal simplex whose ideal vertices have cross ratio . Lobachevsky's function and Clausen's function 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.[1] He was at school with John Galt,[2] who later wrote a biographical essay on Spence.

Analytic structure[edit]

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[edit]

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Particular value identities[edit]

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Special values[edit]

In particle physics[edit]

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[edit]

References[edit]

Further reading[edit]

External links[edit]