Dilogarithm

A special case of the Polylogarithm $\mathop{\rm Li}\nolimits _n(z)$ for $n=2$. It is denoted $\mathop{\rm Li}\nolimits _2(z)$, or sometimes $L_2(z)$, and is defined by the sum

$$\mathop{\rm Li}\nolimits _2(z) = \sum_{k=1}^\infty {z^k\over k^2}$$

or the integral

$$\mathop{\rm Li}\nolimits _2(z) \equiv \int_z^0 {\ln(1-t)\,dt\over t}.$$

There are several remarkable identities involving the Polylogarithm function.

See also Abel's Functional Equation, Polylogarithm, Spence's Integral

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Dilogarithm.'' §27.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1004-1005, 1972.