# Explain this method for integrals (complex analysis)

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 dont 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 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

mfb
Mentor
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.

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 dont 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
Mentor
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.

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 dont understand.

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