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MAT334-2018F => MAT334--Tests => Term Test 2 => Topic started by: Victor Ivrii on November 24, 2018, 05:25:15 AM

Title: TT2B Problem 4
Post by: Victor Ivrii on November 24, 2018, 05:25:15 AM
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 $I$.
Title: Re: TT2B Problem 4
Post by: Zixuan Miao on November 24, 2018, 09:31:44 AM
See attached.

Wrong calculation at that part
Title: Re: TT2B Problem 4
Post by: Zixuan Miao on November 24, 2018, 10:03:59 AM
.
Title: Re: TT2B Problem 4
Post by: Victor Ivrii on November 29, 2018, 09:21:13 AM
Zixuan,
Title: Re: TT2B Problem 4
Post by: hanyu Qi on December 01, 2018, 05:10:08 PM
(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 $$\sqrt i = \frac{\sqrt 2 }{2} + \frac{\sqrt 2 i}{2}$$

$$\ln i = \ln |i| + i arg(i) = i \frac{ \pi }{2}$$
Title: Re: TT2B Problem 4
Post by: Victor Ivrii on December 02, 2018, 12:08:18 PM
After Alex Qi correction from Zixuan post it follows $\newcommand{\Res}{\operatorname{Res}}$

(a) As $R>1$ there is just one singularity inside $\Gamma_{R,\varepsilon}$, namely a simple pole at $z=i$. The residue is
$\Res (\frac{\sqrt{z}\log(z)}{(z^2+1)}, i)= \frac{\sqrt{z}\log(i)}{{2z}}|_{z=i}= \frac{\pi i/2 }{2i}e^{i\pi/4}$ due to the branch selection and therefore $J= 2\pi i \times \frac{\pi i/2 }{2i}e^{i\pi/4}= \frac{\pi^2}{2} e^{3i\pi/4}= \frac{\pi^2}{2\sqrt{2}} (-1-i)$.

(b)
$$\int_{\gamma_R} \frac{\sqrt{z}\log(z)\,dz}{(z^2+1)}|\le \frac{ \pi R \sqrt{R}(\ln(R)+\pi)}{(R-1)^2}\to 0\qquad \text{as }\ \ R\to \infty$$
and
$$\int_{\gamma_\varepsilon} \frac{\sqrt{z}\log(z)\,dz}{(z^2+1)}|\le \frac{ \pi \varepsilon \sqrt{\varepsilon} (|\log \varepsilon|+\pi)}{(1-\varepsilon)^2}\to 0\qquad \text{as }\ \ \varepsilon \to 0$$
(c) In (*) the second integral is $I$ and the first one is
$$\int_{-\infty}^0 \frac{e^{i\pi/2}\sqrt{|x|}(\ln |x|+\pi i)\, dx}{(x^2+1)} = i\int_{\infty}^0 \frac{\sqrt{|x|} (\ln |x|+\pi i) dx}{(x^2+1)}= i I -\pi K$$
with
$$K=\int_0^\infty\frac{\sqrt{|x|} dx}{(x^2+1)}$$
after change of variables. Thus
$$(i+1)I -\pi K= \frac{\pi^2}{2\sqrt{2}} (-1+i).$$
Since both $I,K$ are real
$$I =\frac{\pi^2}{2\sqrt{2}}.$$
As an added value we get $I-\pi K =-\frac{\pi^2}{2\sqrt{2}}$ and $K=\frac{\pi}{\sqrt{2}}$.