Editing Polylogarithm (section)

==Dilogarithm==
{{Main|Dilogarithm}}

The dilogarithm is the polylogarithm of order ''s'' = 2. An alternate integral expression of the dilogarithm for arbitrary complex argument ''z'' is {{harv|Abramowitz|Stegun|1972|loc=§ 27.7}}:
<math display="block">\operatorname{Li}_2 (z) = -\int_0^z{\ln (1-t) \over t} dt = -\int_0^1{\ln (1-zt) \over t} dt.</math>

A source of confusion is that some [[computer algebra system]]s define the dilogarithm as dilog(''z'') = Li<sub>2</sub>(1−''z'').

In the case of real ''z'' ≥ 1 the first integral expression for the dilogarithm can be written as
<math display="block">\operatorname{Li}_2(z) = \frac{\pi^2}{6} - \int_1^z{\ln(t-1) \over t} dt - i\pi \ln z</math>

from which expanding ln(''t''−1) and integrating term by term we obtain

<math display="block">\operatorname{Li}_2(z) = \frac{\pi^2}{3} - \frac{1}{2}(\ln z)^2 - \sum_{k=1}^\infty {1 \over k^2 z^k} - i\pi \ln z \qquad (z \ge 1).</math>

The ''[[Niels Henrik Abel|Abel]] identity'' for the dilogarithm is given by {{harv|Abel|1881}}

<math display="block">\operatorname{Li}_2 \left( \frac{x}{1-y} \right) + \operatorname{Li}_2 \left( \frac{y}{1-x} \right) - \operatorname{Li}_2 \left(\frac{xy}{(1-x)(1-y)} \right) = \operatorname{Li}_2(x) + \operatorname{Li}_2(y) + \ln(1-x) \ln(1-y)</math>

<math display="block">(\operatorname{Re}(x) \le \tfrac{1}{2} \wedge \operatorname{Re}(y) \le \tfrac{1}{2} \vee \operatorname{Im}(x) > 0 \wedge \operatorname{Im}(y) > 0 \vee \operatorname{Im}(x) < 0 \wedge \operatorname{Im}(y) < 0 \vee \ldots).</math>

This is immediately seen to hold for either ''x'' = 0 or ''y'' = 0, and for general arguments is then easily verified by differentiation ∂/∂''x'' ∂/∂''y''. For ''y'' = 1−''x'' the identity reduces to [[Leonhard Euler|Euler]]'s ''reflection formula''
<math display="block">\operatorname{Li}_2 \left(x \right) + \operatorname{Li}_2 \left(1-x\right) = \frac{1}{6} \pi^2 - \ln(x)\ln(1-x) ,</math>
where Li<sub>2</sub>(1) = ζ(2) = <sup>1</sup>⁄<sub>6</sub> ''π''<sup>2</sup> has been used and ''x'' may take any complex value.

In terms of the new variables ''u'' = ''x''/(1−''y''), ''v'' = ''y''/(1−''x'') the Abel identity reads
<math display="block">\operatorname{Li}_2(u) + \operatorname{Li}_2(v) - \operatorname{Li}_2(uv) = \operatorname{Li}_2 \left( \frac{u-uv}{1-uv} \right) + \operatorname{Li}_2 \left( \frac{v-uv}{1-uv} \right) + \ln \left( \frac{1-u}{1-uv} \right) \ln\left( \frac{1-v}{1-uv} \right),</math>
which corresponds to the ''pentagon identity'' given in {{harv|Rogers|1907}}.

From the Abel identity for ''x'' = ''y'' = 1−''z'' and the square relationship we have [[John Landen|Landen]]'s identity
<math display="block">\operatorname{Li}_2(1-z) + \operatorname{Li}_2 \left( 1-\frac{1}{z} \right) = - \frac{1}{2} (\ln z)^2 \qquad (z \not \in ~]-\infty; 0]) ,</math>
and applying the reflection formula to each dilogarithm we find the inversion formula
<math display="block">\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = -\tfrac{1}{6} \pi^2 - \tfrac{1}{2} [\ln(-z)]^2 \qquad (z \not \in [0; 1[) ,</math>

and for real ''z'' ≥ 1 also
<math display="block">\operatorname{Li}_2(z) + \operatorname{Li}_2(1/z) = \tfrac{1}{3} \pi^2 - \tfrac{1}{2} (\ln z)^2 - i\pi \ln z .</math>

Known closed-form evaluations of the dilogarithm at special arguments are collected in the table below. Arguments in the first column are related by reflection ''x'' ↔ 1−''x'' or inversion ''x'' ↔ <sup>1</sup>⁄<sub>''x''</sub> to either ''x'' = 0 or ''x'' = −1; arguments in the third column are all interrelated by these operations.

{{harvtxt|Maximon|2003}} discusses the 17th to 19th century references. The reflection formula was already published by Landen in 1760, prior to its appearance in a 1768 book by Euler {{harv|Maximon|2003|loc=§ 10}}; an equivalent to Abel's identity was already published by [[William Spence (mathematician)|Spence]] in 1809, before Abel wrote his manuscript in 1826 {{harv|Zagier|1989|loc=§ 2}}. The designation ''bilogarithmische Function'' was introduced by [[:sv:Carl Johan Hill|Carl Johan Danielsson Hill]] (professor in Lund, Sweden) in 1828 {{harv|Maximon|2003|loc=§ 10}}. {{harvs | txt | author-link= Don Zagier | first= Don | last= Zagier | year= 1989}} has remarked that the dilogarithm is the only mathematical function possessing a sense of humor.

:{| class="wikitable" style="text-align: center;"
|+ '''Special values of the dilogarithm'''
|-
! <math>x </math>
! <math>\operatorname{Li}_2(x) </math>
! <math>x </math>
! <math>\operatorname{Li}_2(x) </math>
|-
| <math>-1 </math>
| <math>-\tfrac {1}{12} \pi^2 </math>
| <math>-\phi </math>
| <math>-\tfrac {1}{10} \pi^2 - \ln^2 \phi </math>
|-
| <math>0 </math>
| <math>0 </math>
| <math>-1 / \phi </math>
| <math>-\tfrac {1}{15} \pi^2 + \tfrac {1}{2} \ln^2 \phi </math>
|-
| <math>\tfrac {1}{2} </math>
| <math>\tfrac {1}{12} \pi^2 - \tfrac {1}{2} \ln^2 2 </math>
| <math>1 / \phi^2 </math>
| <math>\tfrac {1}{15} \pi^2 - \ln^2 \phi </math>
|-
| <math>1 </math>
| <math>\tfrac {1}{6} \pi^2 </math>
| <math>1 / \phi </math>
| <math>\tfrac {1}{10} \pi^2 - \ln^2 \phi </math>
|-
| <math>2 </math>
| <math>\tfrac {1}{4} \pi^2 - \pi i \ln 2 </math>
| <math>\phi </math>
| <math>\tfrac {11}{15} \pi^2 + \tfrac {1}{2} \ln^2(-1 / \phi) </math>
|-
| 
| 
| <math>\phi^2 </math>
| <math>-\tfrac {11}{15} \pi^2 - \ln^2(-\phi) </math>
|-
|}
:Here <math>\phi = \tfrac{1}{2} (\sqrt{5}+1)</math> denotes the [[golden ratio]].