Integral: $\int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t$

Question

How to prove the following:

$$\int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t=\frac{\pi}{2}\left(\ln(2)+\frac{\pi}{2} i\right)$$

Answer 1

Deform the contour to pick up the residue at $t=i$, $$\begin{eqnarray*} \int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t &=& 2\pi i\, \mathrm{Res}_{t=i} \frac{\ln(t+1)}{t^2+1} \\ &=& 2\pi i \frac{\ln(1+i)}{2i} \\ &=& \frac{\pi}{2}\left(\ln 2+\frac{\pi}{2} i\right). \end{eqnarray*}$$ Here we have assumed the cut starting at $t=-1$ is slightly below the original contour. This appears to be the intended assumption. (If we instead assume the cut is slightly above the original contour we pick up the residue at $t=-i$ and get the complex conjugate of the quoted result.)

Note that the contribution at infinity vanishes since it goes like $\ln(R)/R\,(R\to\infty)$.

Answer 2

Let's evaluate the real and imaginary parts of the integral separately. Using $\Re(\ln(t+1)) = \ln(|t+1|)$ and $\Im(\ln(t+1)) = \pi [t < -1]$, where $[t<1]$ is the Iverson bracket: $$ \begin{eqnarray} \Re \int_{-\infty}^\infty \frac{\ln(t+1)}{t^2+1} \mathrm{d} t &=& \int_{-\infty}^\infty \frac{\ln(|t+1|)}{t^2+1} \mathrm{d} t = \frac{1}{2} \int_{-\infty}^\infty \frac{\ln((t+1)^2 )}{t^2+1} \mathrm{d} t\\ \Im \int_{-\infty}^\infty \frac{\ln(t+1)}{t^2+1} \mathrm{d} t &=& \int_{-\infty}^{-1} \frac{\pi}{t^2+1} \mathrm{d} t \end{eqnarray} $$ The integral for the imaginary part is easy, since $\int \frac{\mathrm{d} t}{t^2+1} = \arctan(t) + C$: $$ \int_{-\infty}^{-1} \frac{1}{t^2+1} \mathrm{d} t = \left.\arctan(t)\right|_{-\infty}^{-1} = \arctan(-1) - \lim_{t\to -\infty} \arctan(t) = -\frac{\pi}{4} - \left(-\frac{\pi}{2}\right) = \frac{\pi}{4} $$ Now to the real part: $$ \int_{-\infty}^\infty \frac{\frac{1}{2} \log((t+1)^2)}{t^2+1} \mathrm{d} t = \int_{-\infty}^\infty \frac{\frac{1}{2} \log(t^2)}{t^2-2t +2} \mathrm{d} t = \int_0^\infty \log(t^2) \frac{1}{2} \left( \frac{1}{(t-1)^2+1}+\frac{1}{(t+1)^2 + 1} \right) \mathrm{d} t = \int_0^\infty \log(t^2) \frac{t^2+2}{t^4+4} \mathrm{d} t \stackrel{u=t^2}{=} \int_0^\infty \frac{\log(u)}{2} \frac{u+2}{u^2+4} \frac{\mathrm{d} u}{\sqrt{u}} = \frac{1}{2} \left.\frac{\mathrm{d} }{\mathrm{d} s} \int_0^\infty u^{s-1} \frac{u+2}{u^2 + 4} \mathrm{d} u \right|_{s=\frac{1}{2}} $$ The latter integral is a sum of Mellin-Barnes transforms of $\frac{1}{u^2+4}$: $$ \int_0^\infty u^{s-1} \frac{u+2}{u^2 + 4} \mathrm{d} u = 2 \int_0^\infty u^{s-1} \frac{1}{u^2 + 4} \mathrm{d} u + \int_0^\infty u^{s} \frac{1}{u^2 + 4} \mathrm{d} u = \frac{2^{s-2} \pi}{\sin\left(\frac{\pi s}{2}\right)} + \frac{2^{s-2} \pi}{\cos\left(\frac{\pi s}{2}\right)} $$ Differentiating, and setting $s=\frac{1}{2}$ yields the result: $$ \int_{-\infty}^\infty \frac{\frac{1}{2} \log((t+1)^2)}{t^2+1} \mathrm{d} t = \pi \log(2) $$ Recombining the real and imaginary parts: $$ \int_{-\infty}^\infty \frac{\ln(t+1)}{t^2+1} \mathrm{d} t = \pi \log(2) + i \frac{\pi^2}{4} $$

Answer 3

It would be easier if you define $$I(\alpha)=\int_{-\infty}^\infty\frac{\ln(\alpha t+1)}{t^2+1}dt.$$ Then \begin{eqnarray*} I'(\alpha)&=&\int_{-\infty}^\infty\frac{t}{(\alpha t+1)(t^2+1)}dt\\ &=&\frac{1}{\alpha^2+1}\int_{-\infty}^\infty\left(-\frac{1}{t+\frac{1}{\alpha}}+\frac{t}{t^2+1}+\frac{\alpha}{t^2+1}\right)dt\\ &=&\frac{1}{\alpha^2+1}\left(-\ln\frac{t+\frac{1}{\alpha}}{\sqrt{t^2+1}}+\alpha\arctan t\right)\big|_{-\infty}^\infty\\ &=&\frac{1}{\alpha^2+1}\left[\ln(-1)+\alpha\pi\right]\\ &=&\frac{\pi}{\alpha^2+1}(i+\alpha), \end{eqnarray*} and hence $I(\alpha)=\pi i\arctan\alpha+\frac{\pi}{2}\ln(\alpha^2+1)+C$. But $I(0)=0$ implies $C=0$. Thus $$I(1)=\pi i\frac{\pi}{4}+\frac{\pi}{2}\ln 2=\frac{\pi}{2}\left(\ln 2+i\frac{\pi}{2}\right).$$

Answer 4

A related problem. Making the change of variables $t=u-1$ yields, $$ \int^\infty_{-\infty}\frac{\ln(t+1)}{t^2+1}\mathrm{d}t = \int _{-\infty }^{0 }\!{\frac {\ln \left( u \right) }{2+{u}^{2}- 2\,u}}{du} + \int _{0 }^{\infty }\!{\frac {\ln \left( u \right) }{2+{u}^{2}- 2\,u}}{du} $$

The last integral has no problem, since $\ln(u)$ is real. For the other integral, we use the substitution $u=-z$, $$ \int _{-\infty }^{0 }\!{\frac {\ln \left( u \right) }{2+{u}^{2}- 2\,u}}{du} = \int _{0}^{\infty }\!{\frac {\ln \left( z \right)+\ln(-1) }{2+{z}^{2}+2\,z}}dz $$ That implies that

$$ \int _{-\infty }^{\infty }\!{\frac {\ln \left( u \right) }{2+{u}^{2}- 2\,u}}{du} = \int _{0 }^{\infty }\!{\frac {\ln \left( u \right) }{2+{u}^{2}- 2\,u}}{du} + \int _{0}^{\infty }\!{\frac {\ln \left( z \right) }{2+{z}^{2}+2\,z}}dz +\int _{0}^{\infty }\!{\frac {\ln(-1) }{2+{z}^{2}+2\,z}}dz$$ $$ = \frac{3\pi\ln(2)}{8} + \frac{\pi\ln(2)}{8}+\ln(-1)\,\frac{\pi}{4} $$ Now using $ \ln(-1)=i\pi $ the result follow $$ \frac{\pi \ln(2)}{2} + \frac{i\pi^2}{4} = \frac{\pi}{2}\left(\ln(2)+\frac{\pi}{2} i\right) $$