Questions on the evaluation of integrals along a locus in the complex plane.

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3answers
30 views

Let C be the circle $|z|=1$. Evaluate $\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$

Let C be the circle $|z|=1$. Evaluate $$\int_{c}\frac{e^{2\pi z}}{(2z+1)^3}dz.$$ Any idea, suggestion, advice or solution.
-1
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0answers
36 views

Contour Integration problem solve

Let C be the circle $z=2+e^{i\theta}$, where $0\le\theta\le2\pi$, Evaluate $$\int_{c}\frac{\sin z}{z^2+2z}dz.$$ i need delicate explanation to understand. i really tried to solve this problem but my ...
-1
votes
0answers
24 views

a poblem about Contour integral [on hold]

Evaluate the integral $\int_{C}\frac{z-1}{z^2+1}dz$ around the circle $|z-2i|=1$. i wanna to know the steps. any idea, suggestion or solution. appreciate your help.
1
vote
1answer
45 views

Calculating $\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x$ using contour integration

I can do this integral using the keyhole contour the answer is:$$\int_0^{\infty } \frac{\ln (x)}{\sqrt{x} \left(a^2+x^2\right)^2} \, \mathrm{d}x = -\frac{\pi (-6 \ln (a)+3 \pi +4)}{8 \sqrt{2}a^{7/2}}$...
1
vote
1answer
89 views

Help me to integrate $\int_{-\infty}^{\infty}{\sin^2{x}\over 1+x^2}dx=2\pi-{\pi^2\over 2}$?

$$I=\int_{-\infty}^{\infty}{\sin^2{x}\over 1+x^2}dx=2\pi-{\pi^2\over 2}$$ Apply residue theorem $$f(x)={\sin^2{x}\over 1+x^2}$$ $$\sin^2{x}={1-\cos(2x)\over 2}$$ $$I={1\over 2}\int_{-\infty}^{\...
3
votes
1answer
45 views

Is this a contour integral question?

I had this in my previous cats that I'm not sure whether it's really a complex analysis question, looks like a differential question with line integrals a bit $$\int_{(1,3)}^{(4,5)} (2y+x^2)\,dx + (...
5
votes
3answers
155 views

How do I prove $\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}{\cos(a-b)}$?

How do I prove these? $$\int_{-\infty}^{\infty}{\sin(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\sin(a-b)}\tag1$$ $$\int_{-\infty}^{\infty}{\cos(x+a)\over (x+b)^2+1}dx={\pi\over e}\color{blue}{\...
1
vote
1answer
35 views

Help, why these are two different results of integral of $\sqrt{z}$ on unit circle depending the choice of Branch cut

everyone, I want test the effect of different choice of branch cut for contour, So I find a simple function, i.e. $\sqrt{z}$ with $z=re^{i\theta}$ on 1st Branch as $$I=\oint_{UnitCircle}{\sqrt{z}dz}$$ ...
0
votes
0answers
38 views

contour integral z/conjugate(z)

I am trying to calculate: $$\int_C \frac{z}{\bar{z}}dz$$ where C is the upper semicircles of the circles centred in (0,0) of radii 1,2 joined at their intersections with the real axis by the real ...
0
votes
3answers
32 views

Simple Contour Integral

I have forgotten much of the complex analysis I once knew. How do I go about using the Cauchy Integral Formula / Residue Theorem to solve this contour integral? The region is the unit circle. $$\...
2
votes
3answers
104 views

Help, Where is wrong when I do same complex integration using two different contours

everyone! please give few hit. I want take the integral $$I=\int_{0}^{\infty}{\frac {dx}{ \sqrt{x}(1+{x}^{2})}} $$ by using the Residue Theorem. I choice two contours in complex plane with $z=r e^{i\...
2
votes
2answers
66 views

Solve this complex integral [closed]

Solve this complex integral $$\lim_{\varepsilon \rightarrow 0} \int_{-\infty}^{\infty} \frac{d\omega}{2\pi i}\frac{e^{-i\omega x}}{\omega + i\varepsilon}$$ Where $\varepsilon > 0$ and $x$ is real....
0
votes
1answer
55 views

Contour Integration, Riemann Zeta (-n)

I was reading Riemann's Zeta Function by H. Edwards, and could not understand the equation on the page 12. \begin{align*} \zeta(-n) &= \frac{\prod(n)}{2\pi i}\int_{+\infty}^{+\infty} \frac{(-x)^{-...
3
votes
1answer
66 views

How to evaluate this Fourier Transform $A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$

This is basically the Fourier transform of a Student´s T pdf. How do we compute it? $$A\int_{-\infty}^{\infty} \frac{e^{ikx}}{(1+x^{2})^{\frac{\nu+1}{2}}}dx$$ for $\nu$ any number greater than zero ...
0
votes
0answers
31 views

$\lim_{\rho\to0}\int_{\gamma_{\rho}}g(z)e^{iz}dz=-\pi i Res(f,a)$ with a pole $a\in\mathbb{R}$

Let $U$ be an open neighbourhood of $\overline{\mathbb{H}}=\{z\in\mathbb{C}:\Im(z)\ge0\}$ and $g:U\rightarrow\mathbb{C}$ meromorphic with a finite number of poles in $\mathbb{H}=\{z\in\mathbb{C}:\Im(z)...
1
vote
2answers
48 views

Closed form and limit of the integral of a rational function

While trying to answer this question, I wondered whether there could be a way to: (A) Find the closed form of the generalization of integrals $I$ and $J$, that is $$I_n=\int_{-\infty}^{+\infty}\frac{...
1
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1answer
42 views

Contour integral $\int_{-\infty}^{\infty}e^{-iax}/(-b+\cos(x))\mathrm dx$ with $a>0$ and $0<b<1$

The integral is $$\text{PV}\int_{-\infty}^{\infty}\frac{e^{-iax}}{(-b+\cos(x))}\, dx$$ with $a>0$ and $0<b<1$. This integral stems from the Fourier transform of a Green's function in ...
22
votes
5answers
1k views

Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$

I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general: $$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$ ...
5
votes
3answers
1k views

Use the Residue Theorem to evaluate the integral:

$$\int_{0}^{∞} \frac{\sqrt{x}}{x^2+2x+5} dx$$ I'm thinking of using the "keyhole" contour, but I'm not sure how to proceed from there. Please help! Thanks!
4
votes
1answer
225 views

A variation of Ahmed's integral

Given that the closed form exist, evaluate the following Integral: $$\displaystyle \int\limits_{0}^{1} \frac{(x^2+4)\sin^{-1}x}{x^4-12x^2+16} \, dx $$
1
vote
3answers
134 views

Evaluate $\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy$ [closed]

How do I evaluate the following integral? $$\int_{-\infty}^{\infty} \frac{(1-\cos { y } )}{\mid{y}\mid^{1+\alpha}}dy=\frac{\pi}{\Gamma(1+\alpha)\sin(\frac{\pi\alpha}{2})}$$ Thank you in advance. ...
1
vote
2answers
58 views

Real integral using a contour integral

I am going to calculate $\int_{-\infty}^{\infty}\dfrac{x \sin \pi x}{x^2+2x+5}dx$ So I have to compute the following limit $\lim_{R \to \infty}\int_{C_1}\dfrac{z \sin \pi z}{z^2+2z+5}dz$ where $C_1$ ...
0
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1answer
31 views

Pdes -theoretical answer

Question.Let $Ω$ be a bounded Connected on $R^3$ with smooth boundary $\partial{Ω}$.Let $u$ be a harmonic function on $Ω$ with continuous derivatives on $Ω\cup \partial{Ω}$ prove that. $$\iint_V \ {\...
0
votes
2answers
96 views

How to solve $\int_{-\infty}^{\infty} \frac {sin(t)}{t^2+1} dt$?

I'm considering here the fact that $$\lim\limits_{R\to\infty} \int_{\Gamma_R} \frac {e^{iz}}{z^2+1} dz=0$$ , where $\Gamma$ is a contour defined as a semicircle centred about the origin, of radius $...
5
votes
0answers
269 views

An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
4
votes
1answer
3k views

Laplace transform of the Bessel function of the first kind

I want to show that $$ \int_{0}^{\infty} J_{n}(bx) e^{-ax} \, dx = \frac{(\sqrt{a^{2}+b^{2}}-a)^{n}}{b^{n}\sqrt{a^{2}+b^{2}}}\ , \quad \ (n \in \mathbb{Z}_{\ge 0} \, , \text{Re}(a) >0 , \, b >0 ...
0
votes
1answer
142 views

Contour integration when pole is outside the contour

Here they are using the pole OUTSIDE the contour? I thought this was illegal according to the residue theorem or we are not supposed to do contour integration with poles outside the contour itself.
0
votes
1answer
27 views

Contour integral independant of parametrisation

I have a question about the definition of contour integrals in $\mathbb{C}$. The same question could be applied to line integrals in $\mathbb{R}^n$ though. $\Gamma \subseteq \mathbb{C}$ is called a ...
2
votes
5answers
96 views

Evaluate $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}$ using contour integration

This question is Exercise 10 of Chapter 3 of Stein and Shakarchi's Complex Analysis. Show that if $a>0$, then $\int_0^{\infty} \frac{\log(x)dx}{x^2+a^2}=\frac{\pi \log(a)}{2a}.$ The hint is ...
2
votes
5answers
70 views

Evaluate the contour integral (Most likely without Green's Theorem)

$\int_{c}\frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$ where $C$ is the triangle with vertices at $(5,5), (-5,5),$ and $(0,-5)$ traversed counterclockwise. (Hint: Be careful about the hypotheses of any ...
2
votes
1answer
20 views

Upper bound for $|\int_\gamma f(z)e^{iz} dz |$

Let $\Omega = \{ z \in \mathbb{C} | Im(z) > 0 \}$ and $f : \Omega \to \mathbb{C}$ a continuous function. We suppose that $\forall z \in \Omega : |f(z)| \leq M|z|^n$ We define $\gamma : [...
1
vote
1answer
43 views

Maximum and minimum modulus principle

Let $U\subset \mathbb C$ be a bounded domain and $f:\overline{U}\to\mathbb C$ continuous and holomorphic $U$. Show that $|f(z)|\leq\max\{|f(w)|:w\in\partial U\}$ for all $z\in U$. Show that ...
6
votes
1answer
110 views

Prove that a complex-valued entire function is identically zero.

Suppose $f$ is entire and $$\iint_\mathbb{C}|f(z)|^2dxdy < \infty$$Prove that $f\equiv 0.$ So far I have: Suppose $f$ is bounded. Then $f$ is constant by virtue of Liouville and so the ...
2
votes
2answers
88 views

Evaluating $\int_c\frac{1}{\sin\frac{1}{z}}\text{d}z$ over $C= \{z\big\vert|z|=\frac{1}{5}\}$

Evaluate $$\int\limits_{|z|=\frac{1}{5}} \frac{1}{\sin\frac{1}{z}}\text{d}z$$ My attempt: I know that this function has non isolated singularity at $0$, and simple poles at $\frac{1}{n \pi}$. ...
3
votes
1answer
159 views

Contour integral for finding $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx$

I can't prove the following result: $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx=\frac{\ln \sqrt{a^2+b^2}}{b}\arctan\frac{b}{a}$ for all $a,b \in \mathbb{R}.$ Well, I consider $\...
1
vote
1answer
558 views

Integral of Meijer G-function

I am trying to integrate this: $$\int_0^\infty \log(1+x^r)x^{a-1}e^{-\beta x}I_v(kx) \ \mathrm dx$$ where $r, a, \beta, v, k$ are arbitrary constants, $v$ is the order of the modified bessel function ...
0
votes
2answers
49 views

Contour Integral finding Poles

I have an integral to solve using appropriate contour integrals; The question is like this. $$\int\limits_0^{2\pi}\frac{\cos(n\theta)d\theta}{1+2p\cos(\theta)+p^2}$$ $$-1<p<1$$ So I thought ...
11
votes
2answers
380 views

Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$

I would like to seek your assistance in computing the sum $$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$ I am stumped by this sum. I have tried summing the residues of $\displaystyle f(z)=\frac{\pi\...
0
votes
1answer
27 views

Complex analysis contour integration calculation check if it is right

Calculate $\int{ \frac{e^z}{z^2(z^2+3)}}dz$ over the rectangle $x=2,x=-2 ,y=2,y=-2$. What i did is find the roots of $z^2+3$ break the $\frac{1}{z^2+3}$ into $\frac{-i/6}{z-3i} +\frac{i/6}{z+3i}$ ...
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vote
1answer
35 views

Contour Integral of $1/(zcos(z))$ around the circle $\lvert z \rvert = n\pi$

This is part of a past exam question from a second year undergraduate complex variable theory course. I am attempting to show that the integral in the title tends to zero as n goes to infinity. The ...
4
votes
1answer
106 views

Using a contour integral about a branch cut to compute $\int \limits ^\infty _0 \frac {\ln x} {x^a (x+1)} dx$

Find the value of $I = \int \limits ^\infty _0 \frac {\ln x} {x^a (x+1)} dx$ for $a \in (0,1)$, placing the branch cut of the logarithm on the positive real axis. You can use the result that $\int \...
0
votes
0answers
19 views

Contour integration problem (not sure if an answer exists)

Here's a tough contour integral question which I haven't been able to solve. It arose out of an amateur fiddling and may very well have a critical error that makes it unable to be evaluated. $f(z)=\...
4
votes
4answers
121 views

Evaluating $\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$ using contour integration

I need your help with this integral: $$\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$$ where $a>0$. I have tried some complex integration methods, but none seems adequate for this ...
3
votes
4answers
221 views

How to find $L = \int_0^1 \frac{dx}{1+{x^8}}$

Let $L = \displaystyle \int_0^1 \frac{dx}{1+{x^8}}$ . Then $L < 1$ $L > 1$ $L < \frac{\pi}{4}$ $L > \frac{\pi}{4}$ I got some idea from this video link. But got stuck while evaluating ...
0
votes
3answers
70 views

Evaluating the real integral $\int_{0}^{2\pi}\frac{1}{2+\sin\theta}d\theta$ using complex analysis

I thought it's value would be zero, since the complex integrand: $$\Im\left(\int_{C}\frac{1}{2+e^{i\theta}}d\theta\right)$$ Where $C$ is the unit disc, is nonsingular. Also $e^{iz}\ne -2$ for any $z$...
7
votes
1answer
927 views

How do you integrate Gaussian integral with contour integration method?

How do you integrate $$\int^{\infty}_{-\infty} e^{-x^2} dx$$ with contour integration method? I do not even know how to setup the problem.
3
votes
3answers
82 views

Evaluating the integral of $\frac{\cos(x) - e^{-x}}{x}$ using contour integration

I am trying to evaluate the value of $$\int_0^\infty\frac{\cos(x) - e^{-x}}{x}dx$$. I am assuming I am supposed to use contour integration, as I was required just before to calculate the value of $$\...
0
votes
1answer
17 views

Contour Integration along a line segment

This may sound like a silly question, but was just wondering if someone can clear this up for me Consider the line segment joining the points a,b. Therefore we have f(t)=a+t(b-a) where t is ...
2
votes
4answers
75 views

Evaluation of the principal value of $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3} \, dx$

I'm trying to evaluate an integral $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3}\,dx$ using Cauchy's theorem. Considering an integral from $-R$ to $-\epsilon$, then a semicircular indentation ...