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

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Two contour integrals in conformal field theory

I am taking a course in conformal field theory. We are treating a Euclidean manifold initially just $\mathbb{R}^2$ with coordinates $(x^1,x^2)$. Next, we define new complex coordinates, $$z=x^1+ix^2, ...
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28 views

Two different results with contour integration

This is probably going to be a stupid question ( I don't feel great today) but I can't get around this problem. $$I = \int_\mathbb R \frac 1 {(3x-2i)^2} dx $$ I thought that using contour ...
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1answer
42 views

How to prove $\lim\limits_{t \to 1^-} \frac{\sqrt{1-t^2}}{2\pi}\int_{S^1}\frac{f(x,y)}{1-tx}ds=f(1,0)$?

$f(x,y)$ is a continuous function defined on unit circle $\ S^1 :$ $x^2+y^2=1$, prove $$\lim\limits_{t \to 1^-} \frac{\sqrt{1-t^2}}{2\pi}\int_{S^1}\frac{f(x,y)}{1-tx}ds=f(1,0)$$ I have tried to ...
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1answer
78 views

Solving this complicated integral using the Residue Theorem

The following is an integral I am trying to evaluate $$I= \int_{-\infty}^\infty f(s) \, ds = \int_{-\infty}^\infty \frac{\frac{1}{(1- \ \ 2 \pi j s )^{m}}-1}{2\pi j s }\ e^{-2\pi j s \ \theta}\ ds ...
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1answer
50 views

Is this contour continuously deformable into a circle?

As an exam question, we had to solve the integral of $\frac{1}{z}$ over the following contour: (The contour is a sequence of straights arcs joining -1, -$\frac{i}{2}$, $\frac{1}{2}$, i, ...
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2answers
46 views

Integral of rational function in the complex plane

Let $P$, and $Q$ be complex polynomials such that $\deg Q \ge \deg(P) + 2$ Prove that there exists $r > 0$ such that if $\gamma$ is a closed curve outside $\{z : |z| \le r\}$, then $$\int ...
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3answers
70 views

How to use complex analysis to find the integral $\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$?

How can I use complex analysis to solve the following: $$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$
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3answers
193 views

Evaluate $\int_1^\infty \frac {dx}{x^3+1}$

I would like some help with the following integral. I would like to find a contour line to evaluate $$\int_1^\infty \frac {dx}{x^3+1}$$ So one can see that on any circumference it goes to $0$, but ...
2
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2answers
48 views

Best way to evaluate integral with contour integration?

I'm trying to evaluate the integral: $$\int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2}dx$$ with contour integration and am not sure if the basic idea of what I'm doing is correct. I know that $$\sin{x} ...
2
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1answer
48 views

Complex Analysis Integrals

I'm unsure how to apply what I've learned in complex analysis to the following question types: $$ \int_{-\pi}^\pi \frac 1 {1 + \sin^2(\theta)}\,d\theta $$ and $$ \int_{-\pi}^\pi \frac ...
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2answers
33 views

Integral along closed contour

In the Laurent series, the coefficient $$b_n = \frac{1}{2\pi i}\int_C\frac{f(z)}{(z - z_0)^{-n + 1}}dz,\qquad\left(\, n = 1,2,\ldots\,\right)$$ collapses to zero when $f(z)$ is analytic in the ...
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0answers
37 views

Contour integral $\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz$ using residues

I am trying to evaluate the contour integral $$\int_{|z|=1}\frac{2z^2+z}{z^2-1}\, dz.$$ In this case the two singular points lie on the boundary (on the contour). So do i count the residues at this ...
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2answers
34 views

Contour integrals with $dx$ instead of $dz$

I was wondering whether a contour integral (over a simple, closed contour) changes if we change the differential to only the axis that contains the singularities. Intuitively, I would think there is ...
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1answer
60 views

Evaluation of $\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta$ with Cauchy's residue Theorem

I have to proof $$\int_0^{2\pi} \frac{1}{1+8\cos^2(\theta)}d\theta = \frac{2\pi}{3}$$ with Cauchy's residue Theorem. I have showed it, but in my solution, there comes $-\frac{2\pi}{3}$. I Show you ...
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1answer
41 views

How can I find the Cauchy Principal Value of this integral using complex analysis?

I'm supposed to solve the real integral using a contour integral (The Cauchy Principal Value). Can someone give me a hand? I cannot seem to be able to do it... This is what I've tried so far: I ...
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1answer
99 views

integration, laurent series, residue therorem

Evaluate the integral $\int_\gamma f(z)dz,$ where $\gamma(t)=e^{it}$, and $0\leqslant t\leqslant2\pi$. For $f(z)$ equal to: $$\dfrac{e^z}{z^3},\quad\dfrac1{z^2\sin z},\quad\tanh ...
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1answer
38 views

Residue theorem with contour integrals

I want to evaluate the integral $$ \int_{\gamma} \frac{1}{z^{2}\sin(z)} dz$$ where $\gamma(t) = e^{it}$ and $ 0 \leq t \leq 2\pi$ using the Residue theorem. I've tried expanding sin(z) with Taylor ...
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3answers
119 views

Evaluate $\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos\theta}\,\mathrm d\theta$

Evaluate $$\int_0^{2\pi} \frac{\sin^2\theta}{5+4\cos(\theta)}\mathrm d\theta$$ This is the final question on my review for my final exam tomorrow, and I will be honest and say that I have no clue ...
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49 views

Evaluate the Cauchy Principal Value of $\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2-2x+2)}dx$

Evaluate the Cauchy Principal Value of $\int_{-\infty}^\infty \frac{\sin x}{x(x^2-2x+2)}dx$ so far, i have deduced that there are poles at $z=0$ and $z=1+i$ if using the upper half plane. I am ...
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1answer
28 views

Generating function of the Laguerre Polynomials

The Laguerre Polynomials have the following integral representations $$L_{n}^{\alpha} (x) = x^{-\alpha} e^x \frac{1}{2\pi i } \oint_c \frac{e^{-z} z^{n+\alpha}}{(z-x)^{n+1}} dz$$ where $c$ is an ...
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1answer
51 views

Countour integral using residue theorem

Evaluate the integral $$ \int_{\gamma} \tanh(z) dz $$ where $\gamma(t) = e^{it}$ and $0 \leq t \leq 2\pi$. I want to do this using the residue theorem but I am unsure of how to work out the poles of ...
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1answer
33 views

Evaluating $\int^{\infty }_{-\infty}\frac {z^3\sin az}{z^4+4}dz$

I'd like to evaluate following integral with contour integration $$\int^{\infty }_{-\infty}\dfrac {z^3\sin az}{z^4+4}dz$$ and I think the best way to solve is to recognize it is equal to the ...
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0answers
20 views

Invert a somewhat tricky characteristic function to find density function

I am interested in find the probability density function corresponding to the characteristic function $\phi(t) = \left(\frac{1 - i b t}{1 - i t}\right)^c$ where $c > 1$ and and $0< b < 1$. ...
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4answers
102 views

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$

Evaluate $\int_{-\infty}^\infty \frac{1}{(x^2+1)^3} dx$ I wasnt exactly sure how to approach this. I saw some similar examples that used Cauchy's theorem.
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2answers
35 views

Poles of $\frac{1}{1+x^4}$

The integral I'd like to solve with contour integration is $\int^{\infty }_{0}\dfrac {dx}{x^{4}+1}$ and I believe the simplest way to do it is using the residue theorem. I know the integrand has four ...
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1answer
17 views

Contour integration over a circle

$$\int_C \frac{\cos(\ z)}{(z)^2} dz$$ where C is any circle enclosing the origin and oriented counter-clockwise. z0 = o of order 2 , f(z) = cos z $$\int_C \frac{\cos(\ z)}{z^2} dz$$ = $2 \pi i ...
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1answer
35 views

evaluating a contour integral where c is $4x^2+y^2=2$

Consider the integral $$\oint_C \frac{\cot(\pi z)}{(z-i)^2} dz,$$ where $C$ is the contour of $4x^2+y^2=2$. The answer seems to be $$2 \pi i\left(\frac{\pi}{\sinh^2 \pi} - \frac{1}{\pi}\right)$$ but ...
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23 views

How do I solve this integral with a branch point at z =0?

The integral $\int_{-\infty}^{\infty}e^{\iota\left(k+\iota\delta\right)x^{2}}dx$ can be written as $\int_{-\infty}^{\infty}\frac{e^{\iota\left(k+\iota\delta\right)z}}{\sqrt{z}}dz$. Here, the branch ...
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2answers
39 views

Evaluate the complex integral [closed]

Evaluate the below integral: $$ \int_{0}^{\infty}{x^{\alpha - 1} \over 1 + x}\,{\rm d}x $$ How to start ?.
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454 views

How to solve $\int_0^{\infty}\frac{\cos{ax}}{x^3+1}dx$?

QUESTION. It is looked for a closed solution for following real integrals $\displaystyle\int_0^{\infty}\displaystyle\frac{\cos{ax}}{x^3+1}dx$ and ...
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1answer
32 views

Find the contour integral around unit circle.

Evaluate the below integral by turning it into a contour integral around a unit circle: $$\int_{0}^{\pi}\frac{\cos2\phi}{1-2a \cos\phi + a^2} d\phi$$ $where\;a\neq \pm1$
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1answer
26 views

evaluate the complex integration

How to evaluate the below integral $$\oint_{c} \frac{dz}{e^{z}-1}$$ where $C$ is the circle $|z|=1$ Can this be done by Cauchy's formula? If yes how? Or do I need to do something else in order to ...
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2answers
28 views

Evaluate the contour integration

Evaluate the below integral: $$\oint_{c}\frac{e^{2z}}{(z+1)^4}dz$$ where $C$ is the circle, $|z|=3$
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1answer
37 views

Evaluate the contour integral

How to evaluate this, $$\oint_{c} \frac{\sin\pi z^2+\cos\pi z^2}{(z-1)(z-2)}dz$$ where $C$ is the circle, $|z|=3$ I tried below things I believe 1 and 2 are simple poles here and the equation can be ...
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0answers
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How to figure out value of L of ML inequality?

Let CR be the upper half of the circle |z|=R(R>2). Show that |Integral CR (zbar^2 + 1)/(z^5 +8) dz| <= (pi R(R^2 +2)) / (R^5 +8) Sorry if this question is dumb. I don't know how to figure out ...
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1answer
25 views

solve a complex integral

I stumbled on this integral, the problem says to solve it with contour integration. Any insights on how to solve this in function of $n$? \begin{equation} \int_{0}^{2\pi}\cos^{2n}(\theta)d\theta ...
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Question about integral over cos^3(theta) on complex plane

I had an integral of $$\int_{0}^{2\pi}\cos^3(\theta) d\theta$$ The answer came out to be integral over the curve $$\int_{C} \dfrac{(z^2+1)^3}{8iz^4} dz$$ $$=-i* \int_{C}\dfrac{(z^2+1)^3}{8z^4} ...
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0answers
56 views

Calculate $\int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx$ using principal branch

I would like to calculate the following integral $$ I = \int_0^3 \frac{x^{\frac{3}{4}} (3-x)^{\frac{1}{4}}}{5-x}\,dx $$ using contour integration but using principal branch of the function, i.e. ...
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Creating contour and gradient map

I have a requirement where, i have been given data set against X, Y Coordinate of a plane. This value is temperature at a point x,y. Now i am suppose to draw a graph with gradient color which displays ...
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2answers
72 views

Evaluaating with $\int_0^\infty \frac{x\sin x}{1+x^2}$ using contour integration?

I'd like to Evaluate $$\int_0^\infty \frac{x\sin x}{1+x^2}$$ The sine function makes the obvious choice $\dfrac{z \sin z}{1+z^2}$ useless since if we integrate over a semicircle sine can become ...
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34 views

How to convert trigonometric integral to a contour integral?

I want to convert $\int_{-\pi}^\pi \frac {d\theta}{(1+sin^2\theta)}$ to a contour integral. I know that I can use the substitution $z=cos\theta + isin\theta = e^{i\theta}$ to get $sin\theta = \frac ...
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1answer
29 views

Contour integration of the bessel function

The Bessel Function $J_n(x)$ is defined, for a natural number $n$ and real number x, as $J_n(x) = \frac{1}{2\pi}\int_0^{2\pi}cos(n\theta-xsin\theta)d\theta.$ By using contour integration with ...
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0answers
47 views

Laplace transform via complex analysis

Let $Y(s) = \frac{2e^{-s}}{s(s^2 + 3s + 2)}$. Then the inverse Laplace transform is \begin{align} y(t) &= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{2e^{s(t - 1)}}{s(s^2 + 3s + ...
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2answers
73 views

Complex integral using cauchy residue formula

I want to compute $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n-1} $ I've proved that $ \displaystyle \int_{0}^{+\infty} \frac{dx}{x^n+1} = \frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}$ in a ...
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35 views

Evaluation of a complex integral, $\int_{z=0}^\infty (c-iz)^{-s-1} e^{-z^{\alpha}}\,dz$

Is there anyone able to solve this integral? $$\int_{z=0}^\infty (c-iz)^{-s-1} e^{-z^{\alpha}}\,dz$$ How can I treat $c-iz$? Both $s$ and $z$ are real numbers. I'm trying to solve it but I don't ...
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1answer
52 views

Contour Integrals Around a Branch Cut

Suppose we have some complex-valued function, $f(z)$ with $\theta_1<\mathrm{arg}(z)\leq \theta_2$ and a branch cut required in the section $[-a,a]$ of the real axis. Now suppose we have a contour, ...
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1answer
51 views

contour integration - ML inequality

I'm trying to show that $$\int_{|z|=r} \frac{\log z}{1+z^2} \ dz $$ goes to 0, as im taking $r \to 0 $ by the ML inequality $$\left| \int_{|z|=r} \frac{\log z}{1+z^2} \ dz \right| \leq \pi r \max ...
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17 views

Contour Integral Around a Circle of Large Radius

I'm given the function $$f(z)=\frac{(z^2-1)^{1/2}}{z^2+1}$$ where $-\pi < arg(z \pm 1) \leq \pi$ and the only branch cut required is the section $[-1,1]$ of the real axis. I'm required, using ...
2
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0answers
31 views

How is half-contour integration possible?

You integrate in a loop around a singularity $z$ and get $2\pi i \text{Res}(z)$. Is there a path of integration such that the result is $\gamma 2\pi i \text{Res}(z)$ with $\gamma\in (0,1)$? If it ...
2
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1answer
28 views

How to compute contour integral?

Use Residue theorem to compute contour integral $$\int_C \frac{4e^z}{\sin z} dz$$ I need help figuring out singularities that are within the circle $|z|= 4$. I am stuck at that part. Thanks in ...