Definition:Spence's Function

Definition

Spence's function is a special case of the polylogarithm, defined for $z \in \C$ by the integral:

$\ds \map {\Li_2} z = -\int_0^z \frac {\map \Ln {1 - t} } t \rd t$

where:

$\ds \int_0^z$ is an integral across the straight line in the complex plane connecting $0$ and $z$

$\Ln$ is the principal branch of the complex natural logarithm.

This article, or a section of it, needs explaining. In particular: Justification is needed to reasssure the readers that the value of the integral is independent of the path. Ultimately it will lead back to Cauchy-Riemann -- but we cannot take this for granted. It would of course be better to use the accepted notation for integration over a countour rather than abuse the real integral notation. But it's a long time since I did any complex integration work, and conventions may have changed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Also known as

Spence's function is also known as the dilogarithm function.

Examples

Example: $\map {\Li_2} {-\phi}$

$\map {\Li_2} {-\phi} = -\dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$

Example: $\map {\Li_2} {-1}$

$\map {\Li_2} {-1} = -\dfrac 1 2 \map \zeta 2$

Example: $\map {\Li_2} {-\dfrac 1 \phi}$

$\map {\Li_2} {-\dfrac 1 \phi} = -\dfrac 2 5 \map \zeta 2 + \dfrac 1 2 \paren {\map \ln \phi}^2$

Example: $\map {\Li_2} 0$

$\map {\Li_2} 0 = 0$

Example: $\map {\Li_2} {1 - \dfrac 1 \phi}$

$\map {\Li_2} {1 - \dfrac 1 \phi} = \dfrac 2 5 \map \zeta 2 - \paren {\map \ln \phi}^2$

Example: $\map {\Li_2} {\dfrac 1 2 }$

$\map {\Li_2} {\dfrac 1 2} = \dfrac 1 2 \paren {\map \zeta 2 - \paren {\map \ln 2}^2}$

Example: $\map {\Li_2} {\dfrac 1 \phi}$

$\map {\Li_2} {\dfrac 1 \phi} = \dfrac 3 5 \map \zeta 2 - \paren {\map \ln \phi}^2$

Example: $\map {\Li_2} 1$

$\map {\Li_2} 1 = \map \zeta 2$

Also see

• Results about Spence's function can be found here.

Source of Name

This entry was named for William Spence.

Sources

• Weisstein, Eric W. "Dilogarithm." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dilogarithm.html