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

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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 ...
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3answers
31 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. $$\...
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3answers
90 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\...
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1answer
53 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)^{-...
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1answer
62 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 ...
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2answers
66 views

Solve this complex integral [on hold]

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....
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$\lim_{\rho\to0}\int_{\gamma_{\rho}}g(z)e^{iz}dz=-\pi i Res(f,a)$ 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)...
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1answer
40 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 ...
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3answers
134 views

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

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. ...
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2answers
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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{...
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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 $C_1$ ...
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1answer
28 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 \ {\...
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2answers
81 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 $...
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5answers
94 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 ...
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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 ...
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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 : [...
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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 ...
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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 ...
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1answer
221 views

A variation of Ahmed's integral

Given that the closed form exist, evaluate the following Integral: $$\displaystyle \int\limits_{0}^{1} \frac{(x^2+4)\sin^{-1}x}{x^4-12x^2+16} \, dx $$
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2answers
87 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 $\frac{1}{n \pi}$. ...
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1answer
104 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 ...
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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 ...
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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 ...
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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 $\frac{-i/6}{z-3i} +\frac{i/6}{z+3i}$ ...
3
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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 $\...
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0answers
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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)=\...
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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 ...
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4answers
215 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 ...
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3answers
69 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 $z$...
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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 ...
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3answers
81 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 $$\...
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2answers
69 views

Prove that: $\frac{2\pi i}{(1 - e^{2i\pi/n})\prod_{k=0, k \neq 1}^{n-1} (e^{i\pi/n} - e^{i(2k-1)\pi/n})} = \frac{\pi/n}{\sin(\pi/n)}$

I am trying to find $\int_0^{\infty} \frac{dx}{1 + x^n}$ using contour integration. I did the computation by taking the contour $[0,R] \cup \gamma_R \cup [R e^{2i\pi/n}, 0]$, with $\gamma_R$ the arc ...
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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 ...
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2answers
44 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
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1answer
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Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
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expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
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152 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
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4answers
58 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral $$\int_{C_R}\...
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1answer
40 views

Calculating Infinite Real Integrals Using Residues

I want to calculate the following real integral using residues and I am unsure how to proceed. $$\int_{-\infty}^{+\infty}\frac{1- x^2}{1+ x^4} dx$$ I know I must change this to a contour integral so ...
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1answer
52 views

What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ is ...
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$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
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3answers
54 views

$\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
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1answer
31 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if $...
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$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R +...
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1answer
29 views

Problem with integration limits using spherical substitution

Good night, i have a problem with this integral, please help me with the integration limits. \begin{align} \int_{-1}^{1}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\int_{-\sqrt{1-x^{2}-y^{2}}}^{\sqrt{1-x^{...
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0answers
35 views

An Alternate proof of Nyquist-Shannon

This problem is from Basic Complex Analysis, Part 2A, by Barry Simon. This problem will provide an alternate proof of the strong from of the Nyquist-Shannon sampling theorem (Theorem 6.6.16 of Part 1)...
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1answer
17 views

Problem with integration limits with cylindric cordinates.

Good night, i have a problem when i go to verify the integration limit $0\leq\theta\leq\varPi/2$ because i think the integration limit go to $0\leq\theta\leq\varPi$ because is an half a circle. $\...
3
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1answer
72 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
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1answer
24 views

Integral estimation

Is it true that $$\int_{-\pi}^{\pi}\left |\frac{iR^{x}e^{i \theta x}}{1-Re^{i\theta}}\right |\,\mathrm{d}\theta \leq 2\pi R\frac{R^{x}}{1-R}\xrightarrow{R\to\infty}0$$ $$\int_{-\pi}^{\pi}\left |\frac{...
4
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0answers
47 views

Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq \frac{\pi}{...