Explain this method for integrals (complex analysis)

[Amad27]

I saw this method of calculating:

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

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 don't understand a few things.

Somehow he makes the relation:

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

How is this established?

And this I don't 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 don't understand what he is exactly doing?
Thanks

 

[mfb]

That relation is not made. The logic:

Consider this function f(x) someone made up: if we integrate it over a square (as described in the comments) the integral is zero. Another way to express the same integral is via its residues: their sum has to be zero.
He then calculates the residues at -2, -1, 0 and the sum of all remaining residues (all at positive integers). The sum of all components is zero. As the sum contains $\sum \frac {2 H_n}{(n+1)(n+2)^3}$ this allows to solve the equation for this expression. You can then use that equation for the original problem.

 

[Amad27]

That relation is not made. The logic:

Consider this function f(x) someone made up: if we integrate it over a square (as described in the comments) the integral is zero. Another way to express the same integral is via its residues: their sum has to be zero.
He then calculates the residues at -2, -1, 0 and the sum of all remaining residues (all at positive integers). The sum of all components is zero. As the sum contains ∑2Hn(n+1)(n+2)3\sum \frac {2 H_n}{(n+1)(n+2)^3} this allows to solve the equation for this expression. You can then use that equation for the original problem.

@mfb, thankyou very much, this was excellent help, physicsforums is great. I would like to ask a few things if you don't mind.

Question 1) He does not define the square though? Where are the vertices?
-I will assume that (-2, -1, 0) are INSIDE the rectangle, not on or outside.

Question 2) How are there infinite singularities at positive integers? http://m.wolframalpha.com/input/?i=digamma(-z)&x=0&y=0
I think I know. \digamma(-z) yields complex infinity for all positive integer values of z.
So there are non removable singularities.

Question 3) How does he get this:

$$\sum_{n=1}^{\infty} Res(f, n) = \sum_{n=1}^{\infty} Res_{z=n} \frac{1}{(z+1)(z+2)^3(z-n)^2} + \frac{2H_n}{(z+1)(z+2)^3(z-n)}$$

More specifically, why does he have (z-n) in the denominator? And where did the $H_n$ appear from?

Thats all for now, hopefully you can guide me, thanks.

 

[mfb]

He does not define the square though?

It is described in the smaller comments below the main post.

I don't know how he got those residuals. Must come from the digamma function.

 

[Amad27]

It is described in the smaller comments below the main post.

I don't know how he got those residuals. Must come from the digamma function.

Okay. mm.. I don't understand.

When he integrated the digamma function, how does the result in a function with H_n?? Thanks