# Inverse of Poly Log function? Asymptotic behavior of Poly Log function?

I am unable to answer important questions such as what is the inverse of PolyLog[3/2,z]? I mean can you express the solution to w = PolyLog[3/2,z] (solve for z in terms of w) in terms of functions known to Mathematica?

Also what does the large z expansion of PolyLog[3/2,z] and PolyLog[5/2,z] look like?

• I do not believe there is a closed form for the inverse of a polylogarithm, but it should not be too hard to construct series expressions: InverseSeries[Series[PolyLog[3/2, x], {x, 0, 5}]] // Simplify. As for asymptotics, have you already seen this? Commented Mar 30, 2019 at 7:47
• Thanks for this. Yes I knew about the wiki. But my comment was in the context of Mathematica. The asymptotic expansion returns unevaluated rather than giving me the wiki answer. Is this also because I am using an outdated version of Mathematica? Commented Mar 30, 2019 at 8:47
• No, I don't think Mathematica is smart enough to know the asymptotic expansion of the polylogarithm (see e.g. this older thread). Commented Mar 30, 2019 at 8:54
• For the second part of your question, look here: functions.wolfram.com/ZetaFunctionsandPolylogarithms/PolyLog/06/… Commented Mar 30, 2019 at 13:54