Toronto Math Forum
MAT3342018F => MAT334Tests => Reading Week Bonussample problems for TT2 => Topic started by: Victor Ivrii on October 29, 2018, 05:19:56 AM

$\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}$
Calculate an improper integral
$$
I=\int_0^\infty \frac{\ln^2(x)\,dx}{(x^2+1)}.
$$
Hint: (a) Calculate
$$
J_{R,\varepsilon} = \int_{\Gamma_{R,\varepsilon}} f(z)\,dz, \qquad f(z):=\frac{\log^2(z)}{(z^2+1)}
$$
where we have chosen the branch of $\log(z)$ such that they are analytic on the upper halfplane $\{z\colon \Im z>0\}$ and is realvalued for $z=x>0$. $\Gamma_{R,\varepsilon}$ is the contour on the figure below:
(b) Prove that $\int_{\gamma_R} f(z)\,dz\to 0$ as $R\to \infty$ and $\int_{\gamma_\varepsilon} f(z)\,dz\to 0$ as $\varepsilon\to 0^+0$ where $\gamma_R$ and $\gamma_\varepsilon$ are large and small semicircles on the picture. This will give you a value of
\begin{equation}
\int_{\infty}^0 f(z)\,dz + \int_0^{\infty} f(z)\,dz.
\label{41}
\end{equation}
(c) Express both integrals using $I$.

As this question is a bit troublesome, there are two scanned pictures. The final answer is π^3/8, which is shown in the last line of the second picture.

Difficult to read. Completely insufficient space between lines.
On test and exam grader could just miss correct elements of the solution.

I think it is OK this time. There are 4 pictures. The first one is for part(a), the second and third one are for part(b), and the last one is for part (c)

Estimate over large semicircle contains fixable errors (I would show on the typed solution) and is overcomplicated: The integrand does not exceed $(R1)^{2}(\ln R +5)^2$ (rough estimate) and the integral does not exceed this multiplied by $\pi R$, so it tends to $0$ as $R\to \infty$
Estimate over large semicircle also contains errors and is overcomplicated: Integrand does not exceed $2\ln \varepsilon^2$ nd the integral does not exceed this multiplied by $\pi
\varepsilon$, , so it tends to $0$ as $\varepsilon\to 0$.