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Spence's function

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Spence's function

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:

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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

In hyperbolic geometry the dilogarithm Li 2 ⁡ ( z ) occurs as the hyperbolic volume of an ideal simplex whose ideal vertices have cross ratio z . 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. He was at school with John Galt, who later wrote a biographical essay on Spence.

Identities

Li 2 ⁡ ( z ) + Li 2 ⁡ ( − z ) = 1 2 Li 2 ⁡ ( z 2 ) Li 2 ⁡ ( 1 − z ) + Li 2 ⁡ ( 1 − 1 z ) = − ln 2 ⁡ z 2 Li 2 ⁡ ( z ) + Li 2 ⁡ ( 1 − z ) = π 2 6 − ln ⁡ z ⋅ ln ⁡ ( 1 − z ) Li 2 ⁡ ( − z ) − Li 2 ⁡ ( 1 − z ) + 1 2 Li 2 ⁡ ( 1 − z 2 ) = − π 2 12 − ln ⁡ z ⋅ ln ⁡ ( z + 1 ) Li 2 ⁡ ( z ) + Li 2 ⁡ ( 1 z ) = − π 2 6 − 1 2 ln 2 ⁡ ( − z )

Particular value identities

Li 2 ⁡ ( 1 3 ) − 1 6 Li 2 ⁡ ( 1 9 ) = π 2 18 − ln 2 ⁡ 3 6 Li 2 ⁡ ( − 1 2 ) + 1 6 Li 2 ⁡ ( 1 9 ) = − π 2 18 + ln ⁡ 2 ⋅ ln ⁡ 3 − ln 2 ⁡ 2 2 − ln 2 ⁡ 3 3 Li 2 ⁡ ( 1 4 ) + 1 3 Li 2 ⁡ ( 1 9 ) = π 2 18 + 2 ln ⁡ 2 ln ⁡ 3 − 2 ln 2 ⁡ 2 − 2 3 ln 2 ⁡ 3 Li 2 ⁡ ( − 1 3 ) − 1 3 Li 2 ⁡ ( 1 9 ) = − π 2 18 + 1 6 ln 2 ⁡ 3 Li 2 ⁡ ( − 1 8 ) + Li 2 ⁡ ( 1 9 ) = − 1 2 ln 2 ⁡ 9 8 36 Li 2 ⁡ ( 1 2 ) − 36 Li 2 ⁡ ( 1 4 ) − 12 Li 2 ⁡ ( 1 8 ) + 6 Li 2 ⁡ ( 1 64 ) = π 2

Special values

Li 2 ⁡ ( − 1 ) = − π 2 12 Li 2 ⁡ ( 0 ) = 0 Li 2 ⁡ ( 1 2 ) = π 2 12 − ln 2 ⁡ 2 2 Li 2 ⁡ ( 1 ) = π 2 6 Li 2 ⁡ ( 2 ) = π 2 4 − i π ln ⁡ 2 Li 2 ⁡ ( − 5 − 1 2 ) = − π 2 15 + 1 2 ln 2 ⁡ 5 − 1 2 Li 2 ⁡ ( − 5 + 1 2 ) = − π 2 10 − ln 2 ⁡ 5 + 1 2 Li 2 ⁡ ( 3 − 5 2 ) = π 2 15 − ln 2 ⁡ 5 − 1 2 Li 2 ⁡ ( 5 − 1 2 ) = π 2 10 − ln 2 ⁡ 5 − 1 2

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:

Φ ⁡ ( x ) = − ∫ 0 x ln ⁡ | 1 − u | u d u = { Li 2 ⁡ ( x ) , x ≤ 1 ; π 2 3 − 1 2 ln 2 ⁡ ( x ) − Li 2 ⁡ ( 1 x ) , x > 1.

References

Spence's function Wikipedia
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