MAT334-2018F > Term Test 2
TT2B Problem 4
Victor Ivrii:
Calculate an improper integral
$$
I=\int_0^\infty \frac{\ln(x)\sqrt{x}\,dx}{(x^2+1)}.
$$
Hint:
(a) Calculate
$$
J_{R,\varepsilon} = \int_{\Gamma_{R,\varepsilon}} f(z)\,dz, \qquad f(z)=\frac{\sqrt{z}\log(z)}{(z^2+1)}
$$
where we have chosen the branches of $\log(z)$ and $\sqrt{z}$ such that they are analytic on the upper half-plane $\{z\colon \Im z>0\}$ and is real-valued for $z=x>0$. $\Gamma_{R,\varepsilon}$ is the contour on the figure below:
(b) Prove that $\int_{\gamma_R} \frac{\sqrt{z}\log(z)\,dz}{(z^2+1)}\to 0$ as $R\to \infty$ and $\int_{\gamma_\varepsilon} \frac{\sqrt{z}\log(z)\,dz}{(z^2+1)}\to 0$ as $\varepsilon\to 0^+0$ where $\gamma_R$ and $\gamma_\varepsilon$ are large and small semi-circles on the picture. This will give you a value of
$$
\int_{-\infty}^0 f(z)\,dz + \int_0^{\infty} f(z)\,dz.
\tag{*}
$$
(c) Express both integrals using $I$.
Zixuan Miao:
See attached.
Wrong calculation at that part
Zixuan Miao:
.
Victor Ivrii:
Zixuan,
please correct
hanyu Qi:
(a)
By residue thrm, $$ \int_{\gamma_{R,\epsilon}} f(z) \text{d}z = 2\pi i Res(f(z),i) = \lim_{z \rightarrow i} (z-i) f(z) = \frac{\ln i \sqrt i}{2i} = \frac{\pi \sqrt i }{4} = \frac{ \sqrt 2 \pi}{8} + i \frac{\sqrt 2 \pi}{8} $$
Since $$ \sqrt i = \frac{\sqrt 2 }{2} + \frac{\sqrt 2 i}{2} $$
$$ \ln i = \ln |i| + i arg(i) = i \frac{ \pi }{2} $$
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