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

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1answer
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Integration of Gamma Function

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanation. This time however I am at this point of integration where: $$\int_{c\ -\ ...
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1answer
25 views

Argument at branch cut

I try to use residue to calculate this integral $$\int_1^2 \frac{\sqrt {(x-1)(2-x)}} {x}\ dx$$ I let $$f(z)=\frac{\sqrt {(z-1)(2-z)}} {z}$$ and evaluate the integral $$\int_{(\Gamma)} f(z)dz$$ along ...
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0answers
35 views

Explanation for summation complex analysis method

This is @Amad27 something happened to my account, which I will get fixed soon, so for now I will ask as a guest until the problem is fixed. Thanks. I saw this method of calculating: $$I = ...
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1answer
64 views

How to find $\max|f(z)|$ in complex analysis?

The $M-L$ estimation lemma inequality states: $$\left |\int_\Gamma f(z) dz\right| < ML(\Gamma)$$ Where $M = \max|f(z)|$ and $L(\Gamma)$ is the arc length of $\Gamma$. Here: Wikipedia: ...
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1answer
52 views

Replacing $\sin(z)$ with $1 - e^{2iz}$

I have seen many integral evaluations within logs where they change the sine to: $$\sin(z) \rightarrow 1 - e^{2iz}$$ Such as here: Contour integral evaluation. I dont understand how those ...
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0answers
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Evaluate $\displaystyle\int_{0}^{1} \frac{\log(x)}{\sqrt{1 - x^2}}$ complex integration [duplicate]

Evaluate: $$2\cdot\int_{0}^{1} \frac{\log(x)}{\sqrt{1 - x^2}} dx$$ Using Complex Integration. I want to do something with the unit circle, but I am not quite sure how to work-around with the unit ...
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3answers
93 views

Integral of $\log(\sin(x))$ using contour integrals

I know the integral is possible with a simple fourier series expansion of $-\log(\sin(x))$ But I am interested in complex analysis, so I want to try this. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
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1answer
37 views

Ml inequality for $\log(z+i)$

I do not need a complete proof, just a hint. This is what the problem is: $$\int_{0}^{\infty} \frac{\log(1+x^2)}{1+x^2} dx$$ Over this contour: The radius is $R$ from the midpoint. I am trying ...
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1answer
58 views

Complex Contour Integrals from integrals from $0 \to 1$

Evaluate: $$\int_{0}^{1} \frac{dx}{1 + x^3}$$ The bounds are not from 0 to infinity or from -infinity to infinity etc.. How can we use complex contour integration for this? Thanks
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2answers
44 views

Finding the value of an integral $\int_{|z|=3}\frac{2z^2-z-2}{z-\omega}dz$

What is the value of $$\int_{|z|=3}\frac{2z^2-z-2}{z-\omega}dz$$ when $|\omega|>3$. I know that when $|\omega|<3$ the value is $2\pi i(2\omega^2-\omega-2)$.
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Estimation Lemma when going to $0$

Here is the problem: Contour Integral problem With help from Jack D'Aurizio We were able to prove that the contour integral of the big semi circle $=0$ as $R \to \infty$. Now the problem is the ...
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2answers
52 views

Integral with contour integration

I want to evaluate the integral: $$\int_{-\infty}^{0}\frac{2x^2-1}{x^4+1}\,dx$$ using contour integration. I re-wrote it as: $\displaystyle \int_{0}^{\infty}\frac{2x^2-1}{x^4+1}\,dx$. I am ...
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1answer
54 views

How to show the contour integral goes to $0$ of semicircle?

Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Image taken and modified from: Complex Analysis Solution (Please Read for background information). $R$ is the big radius, ...
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0answers
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Planning to integrate $\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$ using complex analysis [duplicate]

This is just a plan-out. I want to evaluate: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using a keyhole contour a semi-circle, with base at the x-axis. First I must pick a branch. ...
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2answers
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Transformation of contour integral $\int \frac{z^2}{e^{2\pi i z^3}-1} \operatorname dz$ over the circle $|z|=\sqrt[3]{n+\frac{1}{2}}$

I would like to solve the following: $$\int\limits_{|z|=\sqrt[3]{n+\frac{1}{2}}} \frac{z^2}{e^{2\pi i z^3}-1}\operatorname dz$$ I'm given an hint: "use a transformation $w=z^3$" I would make ...
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0answers
41 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.
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1answer
46 views

How many poles have to be inside the contour?

If we consider $$\int_{0}^{\infty} \frac{dx}{1+x^2}$$ Using complex contour integration only. We choose a contour in the TOP HALF plane. From the poles $z = \pm i$ only, the pole: $z=i$ is ...
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1answer
35 views

Solving an integral using a keyhole based integral (text given)

This is an interesting complex analysis problem; The figure on the bottom left is what is being referred to,Fig7-10. First, lets take a look at the complex line integral. What is the geometry of ...
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1answer
62 views

how to calculate the following integral$\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ [on hold]

calculate the following integral $\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ I need to very hollowing steps.thank you in advance
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1answer
45 views

finding the inverse Laplace transform of $\frac{1}{z\sqrt{z+1}}$

i know that the inverse Laplace transform is given by $$2\pi i \left\{\sum\space\text{ of the residues at the poles of}\space e^{zt}f(z)\right\}- \frac{1}{2 \pi i}\int \text{ along the branch cut}$$ ...
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3answers
139 views

Evaluation of $\int_{0}^{\infty} \cos(x)/(x^2+1)$ using complex analysis.

Evaluate: $$\int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx$$ Using only complex analysis. $$I = \int_{0}^{\infty} \frac{\cos(x)}{x^2 + 1} dx = (\frac{1}{2})\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2 ...
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0answers
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Choosing a contour to integrate over.

What are the guidelines for choosing a contour? For example to integrate a real function with a singularity somewhere. What type of contour from Square, keyhole, circle, etc should be chosen for ...
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2answers
96 views

difficult complex integral $\int_\gamma \frac{1}{z^2+i}dz$

We are asked to calculate $\int_\gamma \frac{1}{z^2+i}dz$ where $\gamma$ is the straight line from $i$ to $-i$ in that direction. My parametrization is simple, I chose $z(t)=i-2it$. Notice that ...
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0answers
14 views

Evaluate the given integral along the given (positively oriented) circle. [closed]

Ok, so I have the following problems that I am working on. It says to evaluate 1) where C is given by |z+1|=1/2 2) where C is given by |z-2|=1/2 3) where C is given by |z|=2 4) where C ...
3
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1answer
77 views

When should I resort to Eulers identity?

I'm working on the following excercise: Calculate: $$\int_0^{+\infty} \frac{x^{\frac{1}{3}}\sin (x+\frac{\pi}{3})}{x^2+1}\operatorname dx$$ Using the contour-integral $\int_{\Gamma} ...
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1answer
29 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
43 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
89 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
47 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
71 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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4answers
224 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 ...
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2answers
49 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
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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 ...
2
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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
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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
100 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
41 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
121 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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2answers
70 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
53 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
35 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 ...
2
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0answers
22 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 $0< b < 1$. ...
2
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4answers
103 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 ...
1
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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 ...