Sunday, 18 September 2016

integration - Closed form for inte0mathrmLi2(lnx),dx?

Inspired by this question and this answer, I decided to investigate the family of integrals
I(k)=e0Lik(lnx)dx,


where Lik(z) represents the polylogarithm of order k and argument z. I(1) evaluates to eγ, but I(2) has resisted my efforts (which can be seen here).





Neither ISC nor WolframAlpha could provide a closed form for its numerical value--however, I've conjectured a possible analytic form. e0Li2(lnx)dx?=3F3(1,1,1;2,2,2;1)+π2(2e5)12+γ22γEi(1)=0.578255559804073275225659054377625577...




Brevan Ellefsen has computed that my conjecture is accurate to at least 150 digits. Brevan also gave the alternate form
π2e6+γG2,01,2(1|10,0)+G3,02,3(1|1,10,0,0).




Is there a closed form for I(2) that doesn't involve Meijer G or hypergeometrics? The simplicity of the following two equations seems to suggest that there might be. (3.1) follows directly from (3), which I've proven here.

k=1I(k)=ek=2I(k)=e(1γ)







PROGRESS UPDATE: Using this equation, I've turned 3F3(1,1,1;2,2,2;1) into
limc1(Ei(1)γc1+1e(c1)2+(1)cΓ(c1)c1+(1)1cΓ(c,1)(c1)2),

but I don't know how to proceed from there. EDIT: This limit leads nowhere. See below.



SECOND PROGRESS UPDATE: After some studying of the properties of the Meijer G function, I've finally cracked the limit; however, the result is an underwhelming 3F3(1,1,1;2,2,2;1). Before I evaluate the limit, I'd first like to state the following intermediate result:





Lemma (4.1): For zC,
G3,02,3(z|1,10,0,0)=γlnz+12ln2(z)z3F3(1,1,1;2,2,2;z)+γ22+π212.




My proof for this can be found here. Now I return to the limit (4).
Consider the following: 1c1=c1(c1)2, (c1)Γ(c1)=Γ(c), and (1)1c=(1)c. Based on these algebraic identities, the limit can be written as
limc1(c1)(Ei(1)γ)+1e+(1)c(Γ(c)Γ(c,1))(c1)2.


In this form, the limit is 00. Using l'Hospital twice, we obtain

limc1(1)c(G4,03,4(1|1,1,10,0,0,c)+Γ(c)(ψ0(c)22iπψ0(c)+ψ1(c)2π22))
=G4,03,4(1|1,1,10,0,0,1)ψ0(1)22+iπψ0(1)ψ1(1)2+π22=G3,02,3(1|1,10,0,0)γ22iγπ+5π212.

Using Lemma (4.1), we know that
G3,02,3(1|1,10,0,0)=3F3(1,1,1;2,2,2;1)+γ22+iγπ5π212,
which can be rewritten as
G3,02,3(1|1,10,0,0)γ22iγπ+5π212=3F3(1,1,1;2,2,2;1).




Thus,
limc1(Ei(1)γc1+1e(c1)2+(1)cΓ(c1)c1+(1)1cΓ(c,1)(c1)2)=3F3(1,1,1;2,2,2;1).


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