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 _3F_3(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 z\in\mathbb{C},
G_{2,3}^{3,0}\left(z\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)=\gamma\ln{z}+\frac12\ln^2(z)-z\,_3F_3(1,1,1;2,2,2;-z)+\frac{\gamma^2}2+\frac{\pi^2}{12}.\tag{4.1}\label{4.1}




My proof for this can be found here. Now I return to the limit \eqref{4}.
Consider the following: \frac{1}{c-1}=\frac{c-1}{(c-1)^2}, (c-1)\Gamma(c-1)=\Gamma(c), and (-1)^{1-c}=-(-1)^{-c}. Based on these algebraic identities, the limit can be written as
\lim_{c\to 1}\frac{(c-1)(\mathrm{Ei}(1)-\gamma)+1-e+(-1)^{-c}(\Gamma(c)-\Gamma(c,-1))}{(c-1)^2}.\tag{4.2}
In this form, the limit is \frac{0}0. Using l'Hospital twice, we obtain

\lim_{c\to 1}{(-1)^{-c}\left(-G_{3,4}^{4,0}\left(-1\left|\begin{array}{c}1,1,1\\0,0,0,c\\\end{array}\right.\right)+\Gamma{(c)}\left(\frac{\psi_0{(c)}^2}2-i\pi\psi_0(c)+\frac{\psi_1(c)}2-\frac{\pi^2}2\right)\right)} \begin{align}&=G_{3,4}^{4,0}\left(-1\left|\begin{array}{c}1,1,1\\0,0,0,1\\\end{array}\right.\right)-\frac{\psi_0(1)^2}2+i\pi\psi_0(1)-\frac{\psi_1(1)}2+\frac{\pi^2}2\\&=G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array} \right.\right)-\frac{\gamma^2}{2}-i\gamma\pi+\frac{5\pi^2}{12}.\tag{4.2a}\end{align}
Using Lemma \eqref{4.1}, we know that
G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)={}_3F_3(1,1,1;2,2,2;1)+\frac{\gamma^2}2+i\gamma\pi-\frac{5\pi^2}{12},\tag{4.3} which can be rewritten as
G_{2,3}^{3,0}\left(-1\left|\begin{array}{c}1,1\\0,0,0\\\end{array}\right.\right)-\frac{\gamma^2}{2}-i\gamma\pi+\frac{5\pi^2}{12}={}_3F_3(1,1,1;2,2,2;1).\tag{4.3a}




Thus,
\eqalign{&\lim_{c\to 1}\left(\frac{\mathrm{Ei}(1)-\gamma}{c-1}+\frac{1-e}{(c-1)^2}+\frac{(-1)^{-c}\,\Gamma(c-1)}{c-1}+\frac{(-1)^{1-c}\,\Gamma(c,-1)}{(c-1)^2}\right)\\=&{}_3F_3(1,1,1;2,2,2;1).\tag{4.4}}


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