# Definition:Spence's Function

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## 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**.

## 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