Method of summing the series with complex analysis

[Amad27]

I asked the same thing on Physicsforums, I am hoping for some feedback here as well.

I saw this method of calculating:


$\displaystyle \displaystyle{I = \int_{0}^{1} \log^2(1-x)\log^2(x) dx}$


http://math.stackexchange.com/questions/959701/evaluate-int1-0-log21-x-log2x-dx


Can you take a look at M.N.C.E.'s method?


I dont understand a few things.


Somehow he makes the relation:


$\displaystyle \displaystyle{\frac{4H_n}{(n+1)(n+2)^3} = \frac{\left( \gamma + \psi(-z) \right)^2}{(z+1)(z+2)^3}}$


How is this established?


And this I dont understand, why did he integrate it,?


And then after he states: "At the positive integers," what is he doing with the residues. I know the residue theorem etc, but I dont understand what he is exactly doing?
Thanks

 

[HallsofIvy]

Do you know what the specific symbols in that equation mean? That is, do you know what "$\displaystyle H_n$", "$\displaystyle \gamma$", "$\displaystyle \psi$" mean?

 

[HLN_Rubi]

Do you know what the specific symbols in that equation mean? That is, do you know what "$\displaystyle H_n$", "$\displaystyle \gamma$", "$\displaystyle \psi$" mean?

H_n is the summation of the harmonic series, 1/1+1/2+1/3+1/4+1/5... +1/n "$\displaystyle \gamma$" is gamma, the limit as you approach infinity of H_n - ln(n). Though I like to use ln(n+1/2) since I can compute it quicker that way. Psi is the function of the d/dx(gamma(x))/gamma(x), aka the derivative of the gamma function over the gamma function, the gamma function being an extension of the factorial function which is given by the integral of (ln(n-1)^x-1 dn with 1 and 0 being the bounds