Editing Polylogarithm (section)

{{Short description|Special mathematical function}}
{{Distinguish|polylogarithmic function|logarithmic integral function}}
{{Use American English|date = March 2019}}

In [[mathematics]], the '''polylogarithm''' (also known as '''Jonquière's function''', for Alfred Jonquière) is a [[special function]] {{math|Li{{sub|''s''}}(''z'')}} of order {{mvar|s}} and argument {{mvar|z}}. Only for special values of {{mvar|s}} does the polylogarithm reduce to an [[elementary function]] such as the [[natural logarithm]] or a [[rational function]]. In [[quantum statistics]], the polylogarithm function appears as the closed form of [[integral]]s of the [[Fermi–Dirac distribution]] and the [[Bose–Einstein distribution]], and is also known as the '''Fermi–Dirac integral''' or the '''Bose–Einstein integral'''. In [[quantum electrodynamics]], polylogarithms of positive [[integer]] order arise in the calculation of processes represented by higher-order [[Feynman diagram]]s.

The polylogarithm function is equivalent to the [[Hurwitz zeta function]] — either [[Function (mathematics)|function]] can be expressed in terms of the other — and both functions are special cases of the [[Lerch transcendent]]. Polylogarithms should not be confused with [[polylogarithmic function]]s, nor with the [[offset logarithmic integral]] {{math|Li(''z'')}}, which has the same notation without the subscript.

<gallery mode="packed" heights="140px" caption="Different polylogarithm functions in the complex plane">
File:Complex polylogminus3.jpg|{{math|Li{{hairsp}}{{sub|–3}}(''z'')}}
File:Complex polylogminus2.jpg|{{math|Li{{hairsp}}{{sub|–2}}(''z'')}}
File:Complex polylogminus1.jpg|{{math|Li{{hairsp}}{{sub|–1}}(''z'')}}
File:Complex polylog0.jpg|{{math|Li{{sub|0}}(''z'')}}
File:Complex polylog1.jpg|{{math|Li{{sub|1}}(''z'')}}
File:Complex polylog2.jpg|{{math|Li{{sub|2}}(''z'')}}
File:Complex polylog3.jpg|{{math|Li{{sub|3}}(''z'')}}
</gallery>
The polylogarithm function is defined by a [[power series]] in {{mvar|z}}, which is also a [[Dirichlet series]] in {{mvar|s}}:
<math display="block">\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s} = z + {z^2 \over 2^s} + {z^3 \over 3^s} + \cdots</math>

This definition is valid for arbitrary [[Complex numbers|complex]] order {{mvar|s}} and for all complex arguments {{mvar|z}} with {{math|{{abs|''z''}} < 1}}; it can be extended to {{math|{{abs|''z''}} ≥ 1}} by the process of [[analytic continuation]]. (Here the denominator {{mvar|k{{sup|s}}}} is understood as {{math|exp(''s'' ln ''k'')}}). The special case {{math|1=''s'' = 1}} involves the ordinary [[natural logarithm]], {{math|1=Li{{sub|1}}(''z'') = −ln(1−''z'')}}, while the special cases {{math|1=''s'' = 2}} and {{math|1=''s'' = 3}} are called the [[dilogarithm]] (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated [[indefinite integral|integral]] of itself:
<math display="block">\operatorname{Li}_{s+1}(z) = \int_0^z \frac{\operatorname{Li}_s(t)}{t} dt</math>
thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders {{mvar|s}}, the polylogarithm is a [[rational function]].