How to calculate the integral $\int_0^{+\infty} \frac{\ln{x}}{(x+1)(x^2+1)} dx$ using residues?

Question

I tried to solve $\int_0^{+\infty} \frac{\ln{x}}{(x+1)(x^2+1)} dx$ by calculating the integral $\int_0^{+\infty} \frac{\ln^2{x}}{(x+1)(x^2+1)} dx$.

At first I precalculated the integral $\int_C \frac{\ln^2{z}}{(z+1)(z^2+1)} dx$, where $C$ is the contour like on the image:

So,

$$\int_C \frac{\ln^2{z}}{(z+1)(z^2+1)} dz = \int_{B(0,\epsilon)} \frac{\ln^2{z}}{(z+1)(z^2+1)} dx +$$ $$+ \int_{R_{up}^+} \frac{\ln^2{z}}{(z+1)(z^2+1)} dz + \int_{B(0,R)} \frac{\ln^2{z}}{(z+1)(z^2+1)} dz + \int_{R_{down}^+} \frac{\ln^2{z}}{(z+1)(z^2+1)} dz =$$ $$= 0 + \int_0^{+\infty} \frac{\ln^2{x}}{(x+1)(x^2+1)} dx + 0 + \int_{+\infty}^0 \frac{(\ln{x} + 2i\pi)^2}{(x+1)(x^2+1)} dx =$$ $$= - 4i\pi \int_0^{+\infty} \frac{\ln{x}}{(x+1)(x^2+1)} dx + 4\pi^2 \int_0^{+\infty} \frac{1}{(x+1)(x^2+1)} dx$$

But according to the Cauchy Residue Theorem,

$$\int_C \frac{\ln^2{z}}{(z+1)(z^2+1)} dz = 2i\pi (res_{z=-1} f(z) + res_{z=-i} f(z) + res_{z=i} f(z)) =$$ $$= 2i\pi \Big( \frac{\ln^2(-1)}{(-1)^2+1} + \frac{\ln^2(-i)}{(-i+1)(-i-i)} + \frac{\ln^2{i}}{(i+1)(i+i)} \Big) =$$ $$= 2i\pi \cdot \Big( -\frac{3\pi^2}{8} \Big) = -\frac{3i\pi^3}{4}$$

Since $\int_0^{+\infty} \frac{1}{(x+1)(x^2+1)} dx = \frac{\pi}{4}$, I can find the answer on my question:

$$\int_0^{+\infty} \frac{\ln{x}}{(x+1)(x^2+1)} dx = \frac{4\pi^2 \cdot \frac{\pi}{4} + \frac{3i\pi^3}{4}}{4i\pi} = \frac{\pi^2}{4i} + \frac{3\pi^2}{16} = - \frac{i\pi^2}{4} + \frac{3\pi^2}{16}$$

But I know that the correct answer is $- \frac{\pi^2}{16}$. It seems very strange that I have imaginary part left. I've tried to find solutions for the same integrals, but they just use keyhole contour as I do and it's okay. Can you help me to find a mistake?

Answer 1

This is due to the problem of branch cuts. You should take

$$\ln(-i)=\frac{3\pi i}2\qquad \ln(i)=\frac{\pi i}2$$

Here is the reason why:

To apply residue theorem, the contour must not enclose any branch cuts. As you have chosen a contour that avoids the non-negative real axis, this forces you to take the branch cut of logarithm along $[0,\infty)$.

This choice of branch cut corresponds to $\arg z\in [2n\pi,2(n+1)\pi)$, for any $n\in\mathbb Z$.

When you write $$\int_{R^+_{up}}\frac{\ln z}{p(z)}dz=\int^\infty_0\frac{\ln x}{p(x)}dx$$ you implicitly assumed that $\lim_{\epsilon\to 0^+}\ln (x+i\epsilon)$ is real for $x>0$, or equivalently in short, $\ln z$ is real on the upper positive real axis.

Since $\lim_{\epsilon\to 0^+}\ln (x+i\epsilon) =\ln x+2n\pi i$, to make it real we must choose $n=0\implies\arg z\in [0,2\pi)$.

(In case you wonder: branch cut actually marks the discontinuities of a function. When a function is discontinuous along the positive real axis, intuitively it is equivalent to breaking the positive real axis into upper and lower parts, and the function takes different values on these two parts.)

In essence, the imaginary part of $\ln z\equiv \ln |z|+i\arg z$ must be in the range $[0,2\pi)$. This answers your question.

(This is an image of the Riemann surface of the principal logarithm, i.e. $\arg z\in[-\pi,\pi)$. This logarithm is discontinuous along the negative real axis. As you can see, the negative axis is broken into two parts: the white line at the edge of green sheet and the white line on the orange sheet. For continuous functions, such splitting/breaking never occurs.)