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Questions tagged [contour-integration]

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

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Integration of $\frac{e^{ibx}}{\sin(ax)^2}$ [on hold]

I try to solve this integral , $\int \frac{e^{ibx}}{\sin(ax)^2}dx$ where, $a$ and $b$ are real. Do you have ideas on how to proceed ?
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1answer
28 views

$\int_{\Gamma} \dfrac{z^2+z^{-2}}{(z^*-r_1)(r_2-z^*)}dz$

Problem:: $\int_{\Gamma} \dfrac{z^2+z^{-2}}{(z^*-r_1)(r_2-z^*)}dz$ where $\Gamma= \{ z: |z|=r \}$ ($r_2>r>r_1>0$). Question: How to solve this integral? My attempt: My first idea is to use ...
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30 views

Contour integral over a non-rectifiable contour

In chapter 16 of Mathematical Analysis Apostol defines contour integrals in terms of Riemann-Stieltjes integrals, specifically: $$\int_\gamma f = \int_a^b f[\gamma(t)]\,d\gamma(t)$$ whenever the ...
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1answer
54 views

Closed from of $\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$?

I've been trying to find the general form of a certain group of integrals of the form$$I(a,n)=\int_0^{\infty} \frac{e^{iax}}{x^{n}+1}dx$$ I know that the real part of $I(a,2)$ can be calculated using ...
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1answer
54 views

Showing that the limit of contour integrals is zero

I want to prove that $$\lim\limits_{N \to \infty}{\oint_{C_N}{\frac{z}{\exp(z)-1}}\cdot\frac{dz}{z^{2\cdot k+1}}}=0,$$ where $C_N=\{z\in \mathbb C : |z|=2\pi(N+\frac{1}{2}) \}$ and $k\in \mathbb ...
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35 views

Use The Fundamental Theorem of Contour Integration or otherwise to evaluate the following integrals. (If it can't be used state why) [closed]

a) $\int|z|dz$, where $\gamma(t) = 3e^{it} (0 \le t \le \pi)$; b) $\int \cos z - z\sin z dz$, where $\gamma(t) = (−1 + 2t) + it (0 \le t \le 1).$
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Stuck on proof using Cauchy's integral formula

I posted my attempted proof to this question here but I realized that I was wrong in taking the limit, and that the proof did not make sense. So I am still stuck on this problem let $f: \Omega \...
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How to integrate $\int_0^{+\infty} \frac{x^2\ln x}{x^4-x^3+1}\,\mathrm{d}x$?

First I think it may use the method of contour integration,but I have no idea how to create a contour.And is there some normal method to solve it? $$\int_0^{+\infty} \frac{x^2\ln x}{x^4-x^3+1}\,\...
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Stuck on contour integral, can I use Cauchy's theorem instead?

I am trying to calculate $$\int_{|z-2| = 3} e^{1/z}$$ I parametric the circle of radius three centered at 2 by $\gamma(t) = 3e^{it} +2$ and so I can instead evaluate $$\int_\gamma f(\gamma(t))\...
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24 views

Cauchy Principal Value calculation

I am self-studying the residue theorem and its applications and I tried solving a problem which involves finding the principal value for an improper integral but I am not sure if my approach/answer is ...
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1answer
35 views

Is the residue theorem the correct approach for this integral?

I want to calculate $$\int_{\gamma} \frac{\sin z}{z^2 + 1}dz$$ where $\gamma$ is the upper-half circle of radius 2 centered at the origin starting at 2. I know that since $\gamma$ is not a closed ...
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1answer
40 views

contour integral with given contour

I know that there are three simple poles e^(pii/3),e^(pii),e^(5pi*i/3) and Res(1/(x^3+1))=Res(1/3x^2)=Res(-x/3) but i dont know how to deal with that kind of contour I gussed e^(pi*i/3) is only ...
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Integration problem (contour?) [closed]

I'm trying to solve $$A_2\int_{0}^{1}\frac{A\sqrt{t(1-t^2)}}{2(t^2-1)}e^{zt}dt = \frac{e^{z}}{\sqrt{z}},$$ where $A_2$ is a constant. Would Watson's lemma apply here?
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Simple contour integration problem

I have the differential equation $2zw'''+5w''-2zw'-3w=0$ to which I pick an ansatz of the form $w(z) =\int_{C}^{} P(t)e^{zt} dt.$ Solving, I find $P(t) = \frac{A\sqrt{t(1-t^2)}}{2(t^2-1)}.$ For $...
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how to integrate $\frac{-1+e^{-i k x}}{x^2}$

How do I integrate the following?$$\int_{-\infty }^{\infty } \frac{-1+e^{-i k x}}{x^2} \, dx$$ I am not very familiar with complex analysis, but I did try to use contour integral to do this but I ...
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2answers
64 views

Evaluate $\int_{-\infty}^\infty \frac{1}{\sqrt{z^2 + 1}}\frac{1}{z - \alpha} dz$.

Evaluate $$\int_{-\infty}^\infty \frac{1}{\sqrt{z^2 + 1}}\frac{1}{z - \alpha} dz\,.$$ What is an elegant way to evaluate this integral for Im $\alpha >0$? I imagine using residue theorem will lead ...
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3answers
81 views

Using the residue theorem to evaluate an integral

$$I = \int_0^{\infty} \frac {dx}{x^6+1}$$ My thinking is that I can use the property of even functions to integrate across the whole domain from negative infinity to positive infinity. Can the ...
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1answer
52 views

Integral from superconductivity theory

I am dealing with the following integral $$\int_{0}^{\infty}\frac{dx}{x^2}\left(\frac{1}{\cosh^2x}-\frac{\tanh{x}}{x}\right).$$ My attempt to calculate this integral: calculate residues of $$f(x)=\...
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can solve this by contour integration?

how is done this integral: $$\int_{0}^{2 \pi} \ln(z-Re^{i \theta})e^{i \theta}d\theta$$ I try by substitution of $u=z-Re^{i \theta}$ and i get 0 but i don't know if is correct, perhaps by contour ...
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1answer
47 views

Shifting integration in direction of a cone

While reading through a (simple) complex analysis paper, I came across the following type of argument: Let $C$ be an open cone and $f\colon \mathbb C^n \to \mathbb C$ be a function of $n$ complex ...
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Laplace transform, bromwich contour - basic question

I am confused about the following two observations which seem contradictory: It is stated that the region of convergence of the Laplace transform is a half space. That is $L(s)$ is defined for all $...
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2answers
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Evaluate $\int_0^{\infty}\frac{\ln x}{x^a(x+1)}dx$ where $0<a<1$

I'm trying to compute this integral, $$\int_{0}^{\infty}\frac{\ln x}{x^{a}(x+1)}dx \hbox{ where } 0<a<1$$ I drew a typical Pacman contour with branch cut at positive real axis. Then, we have $$\...
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Integral of a complex function over contour (continued)

I'd like to double check that my method to evaluating the following integral over a contour is correct. I asked how to go about integrating over such a contour in Integral of a complex function over ...
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2answers
80 views

Integral of a complex function over contour

I'd like to double check that my method to evaluating the following integral(s) over a contour is correct: $\Gamma$ is a simple closed contour given by the path moving from 0 to 1 along the real axis,...
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2answers
40 views

Integral of complex number over a contour

I have $$\int_{-1}^1 |z|dz$$ I need to calculate the integral where the integration contour is the upper semi-circle with unit radius. I calculated the integral in $(-1; 1)$ section; the answer is 1, ...
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1answer
57 views

Limiting value of a contour integral

Fix an integer $r \geq 1$, and a complex real $u$. (edit: I think $u$ is meant to be real) For an integer $k \geq 1$ define $$ a_k = \int_{ |z| = (2k+1)\pi } \frac{z \exp( u z)}{(e^z - 1)z^{2r+1}}dz. ...
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Conjugate of contour integrals

Let $\varphi : \{ z \in C | |z| = 1\} \rightarrow C$ be a continuous function and let $\gamma$ be the circular contour centered at 0 of radius 1 positively oriented. Prove that $$\overline{\int_{\...
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Why is the contour integral in upper plane different from the lower plane in this case?

Why is the contour integral in upper plane different from the lower plane in this case? $\int_{-\infty}^{\infty}\mathrm{d}k\frac{1}{(k+a)(k-a)(p-k-b)(p-k+b)}$ where Im[a] and Im[b] are negative and ...
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2answers
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Contour intergal of a rational trigonometric function, can't find my mistake.

This is the integral : $$I = \int_{0}^{2\pi} \frac{dx}{(5+4\cos x)^2}\ $$ Which, according to wolfram alpha, should evaluate to $\frac{10\pi}{27} $, but the value i find is $\frac{20\pi}{27} $. These ...
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Solutions check. Find $\int_{-b}^{b} \Big( \frac{1}{t + ix} - \frac{1}{t - ix} \Big) dt$

Let $x > 0$. Find $$\lim_{b \to \infty} \int_{-b}^{b} \Big( \frac{1}{t + ix} - \frac{1}{t - ix} \Big) dt$$ I solved this problem but I am not quite confident about my steps. So please, could you ...
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1answer
86 views

Contour integration - complex analysis

I am trying to solve the following integral using a contour (large semi-circle connected to smaller semi-circle in the upper-half plane): $$\int_0^{\infty} \frac{\log^4(x)}{1+x^2} dx.$$ I have ...
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1answer
64 views

How to find the Fourier transform of $\frac{(t^2+2)}{(t^4+4)}$ and use it to evaluate $\frac{(t^2+2)^2}{(t^4+4)^2}$?

I'm asked to evaluate the Fourier transform of $\dfrac{(t^2+2)}{(t^4+4)}$, and then use it to evaluate the integral from minus infinity to plus infinity of $\dfrac{(t^2+2)^2}{(t^4+4)^2}$. Part 1: ...
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Integral which might require contour integration

For a physics problem I need to calculate the following integral: $$I = \int \frac{(\alpha^2/k^2)}{\frac{\vec{p}^2}{2}-\frac{(\vec{p} - \vec{k})^2}{2} -1 +i\varepsilon} \frac{d^3 \vec{k}}{(2\pi)^3} $...
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2answers
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Why is integral not equal to zero even though the path is closed?

I have an integral $$\int{z^{i}}dz$$ the path is $e^{it}$ where $t$ is between $0$ and $2\pi$. Why is it not equal to zero even though the path is closed.
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Complex line integrals

Suppose we have an analytic function then Why complex integral of that function does not depend on the path of integration?
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A tricky contour integral.

Let $f$ be holomorphic on a domain $D$ containing $\overline{D}(0,1)$, and $\gamma$ be the positively oriented circular contour centred at 0 of radius 1. Prove that $$ \frac{1}{2 \pi i} \int_{\gamma} \...
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Evaluation of contour integration help involving exponential and cosh$z$

Let the contour $\gamma$ be a positively oriented cifrcle of radius 2 centered at zero, traversed once. Evaluate I = $\int_{\gamma}$ $\frac{dz}{(1-e^{iz})cosh(z)}$. This is what Ive done so far ...
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2answers
52 views

Evaluating a simple integral with the Cauchy residue theorem and a semicircular contour

I am taking a course next week that requires some basic integral techniques from complex analysis and I've been trying to quickly teach it to myself. I was given this sample problem to test my ...
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1answer
59 views

Evaluate $\int_\gamma\frac{1}{z}dz$ over a circle outside the origin

I am trying to do the evaluate this integral: $$\int_{\gamma} \frac{1}{z} dz$$ where $\gamma$ is a circle that does not contain the origin on itself either on its inside, I mean, $\gamma$ is a ...
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Show a Coarea-formular-like identity

I want to show that $\displaystyle u(x)=\frac{1}{|B_r(x)|}\int_{B_r(x)}u\,\mathrm{d}x = \frac{1}{|\partial B_r(x)|}\int_{\partial B_r(x)}u\,\mathrm{d}S$ where $B_r(x)$ is the ball with radius $r$ ...
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Branch cut for $\sqrt{z^2-a^2}$ in the lower half of the complex plane (instead of $[-a,a]$) - How does this change the function?

I have given a function $\sqrt{z^2-a^2}$ with $a>0$. At first i have chosen the branch cut on the real axis at $-a<z<a$: $$f(z)=\sqrt{z-a}\sqrt{z+a}=\sqrt{|z-a||z+a|} \exp({i\frac{\theta_1+\...
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Contour integral with branch points inside argument of logarithm

This question comes from the context of calculating the grand potential for a simple toy problem (a linear chain of masses connected by springs with a mass defect) using statistical field theory. In ...
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1answer
23 views

Considering the work done from two different paths

I am to consider two paths traveled below: With the following vector field: $$\vec F = \frac{-y \hat x + x\hat y}{x^2+y^2}$$ And I am to consider the work done going from $(-1,0)$ to $(1,0)$ with ...
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The convergence of an improper integral

Let us find all values of $\alpha$ such that the following improper integral converges: $$I(\alpha):=\int_{1}^{\infty} \frac{ e^{\dot{\imath}(x^2-2x)}}{x^{\alpha}}\,dx$$. When $\alpha>1$, $I(\...
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2answers
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Contour integration - evaluating $\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+1)} \ dx$ around a semi-circle

I am trying to show that $$\int_{-\infty}^{\infty}\frac{\sin(x)}{x(x^2+1)} \ dx=\pi(1-e^{-1}).$$ I considered a semi-circle in the upper-half plane, indented at the origin, orientated counter-...
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Show that a function is analytic on the punctured complex plane

I have to show that show that $\sum_{k = 1}^{\infty} \frac{1}{n! z^n}$ is analytic on $\mathbb{C}\{0\}$ and calculate its integral around the unit circle. My attempt is to try and use the analytic ...
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if $f_n(z)\rightarrow f(z)$ uniformly, then $ \frac{f'_n(z)}{f_n(z)}\rightarrow\frac{f'(z)}{f(z)}$ uniformly?

I am reading "The Taylor Series: An Introduction to the Theory of Functions of a Complex Variable", Dienes P.N., Dover (1957). In the proof of a theorem (Hurwitz) on page 351 it says that, if $f_n(...
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75 views

Does the integral $\int_{0}^{\infty} \frac{x^3\,\cos{(x^2-x)}}{1+x^2}$ diverge

Does the integral $$J:=\int_{0}^{\infty} \frac{x^3\,\cos{(x^2-x)}}{1+x^2}dx $$ diverge ? If we integrate by parts we find $$J=\lim_{a\rightarrow +\infty} \frac{a^3}{(1+a^2)(2a-1)}\cos{(a^2-a)} -\\ \...
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2answers
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Tricky Indented Contour Integral of $\int_{0}^{\infty} \frac{x^{3/4}}{(x^2+1)^2} \ dx$

I am trying to evaluate the integral $$\int_{0}^{\infty} \frac{x^{3/4}}{(x^2+1)^2} \ dx,$$ by finding a suitable branch of $z^{3/4}$ and integrating the function $z\rightarrow z^{3/4}/(z^2+1)^2$ ...
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23 views

Contour integral over a square

$f(z) = \frac{z^2e^z}{2z+i}$, and the contour is the square with vertices at $1+i, -1+i, -1-i,$ and $1-i.$ The issue I'm having with this integral is that the singular point $z = -\frac{i}{2}$ is ...