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

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2
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1answer
30 views

Contour for calculating this complex integral

As part of the proof of the Prime Number Theorem in my online notes, we are told to show the following identity: For $y>0$, $c>0$ show that $ \int _{c-i \infty}^{c+i \infty} ...
0
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2answers
83 views

$\int_{0}^{\infty}\frac{\sqrt{x}}{1+x^2}\,dx$ [duplicate]

I want to evaluate the integral $\displaystyle \int_{0}^{\infty}\frac{\sqrt{x}}{1+x^2}\,dx$ using complex analysis methods. I know that I have to use a keyhole contour, but I don't know which function ...
4
votes
2answers
88 views

On the value of $e^{ix}$ at $\pm \infty$

Consider the integral $$ \int_{-\infty}^{+\infty} e^{ix} \, dx.$$ Integrating, we have $$\left[-ie^{ix}\vphantom{\frac11}\right]_{-\infty}^{+\infty},$$ and we need to evaluate the limits of $e^{ix}$ ...
4
votes
1answer
55 views

Calculating a contour integral

I want to evaluate the integral $$\int_{\gamma} \sin{(2z)} \ {\rm d}z$$ where $\gamma$ is the line segment joining the point $i+1$ to the point $-i$. Thus $\gamma(t) = -i+t(2i+1)$ for $0\le t\le1$. ...
0
votes
2answers
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Evaluate $\oint_c {4z - 1}\,dz$ along the circle $|z| = 1$

Evaluate $\displaystyle\oint_c {4z - 1}\,dz$ along the circle $|z| = 1$ from the point $(0,-1)$ to $(1,0)$ My question is how to do a contour integration in the circle? I only know to do it in ...
3
votes
1answer
86 views

Evaluating an alternating sum using contour integrals

Evaluate: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{3n-1}$$ Using contour integration. Normally I would use $\pi\csc(\pi z)f(z)$ and evaluate the residue multiply by (-1) and divide by $2$ if the ...
1
vote
2answers
51 views

Countour integral $\int {{{(\overline z )}^2}dz} $

Evaluate $\int {{{(\overline z )}^2}dz} $ along the straight line segment from $z=0$ to $z=2+i$. My attempt to this question is I change z into $x+iy$ and do the integration; $$\int_0^{2 + jy} ...
2
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1answer
109 views

Evaluating$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $using residues

I need help to solve the next improper integral using complex analysis: $$ \int_{-\infty}^{\infty} \frac{x^6}{(4+x^4)^2} dx $$ I have problems when I try to find residues for the function $ f = ...
1
vote
1answer
64 views

What kind of contour (if any) can be used for these types of trigonometric integrals?

I've encountered the following integral while trying my hand at differentiating under the integral sign: $$-\int_{-\pi}^\pi\frac{x\sin ax}{2+\cos ax}\,dx$$ and I remember seeing something similar from ...
3
votes
1answer
129 views

Proving an Integral with Cauchy Residue Theorem

I need help proving this. The clue given is that Cauchy residue theorem can be used: $${1 \over {2\pi j}}\int_{c\ -\ j\infty}^{c\ +\ j\infty} x^{-s}\sigma^{s-1} ...
3
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1answer
76 views

Integration with Beta Function $\beta$ [closed]

Given that: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty}\left({\sigma\,x^{-1}}\right)^u\beta\left(u,a\right)du=\left(1-{x \over \sigma}\right)^{a-1}$$ whereby $\sigma>0$, $a>0$ and $x$ is a real ...
4
votes
1answer
71 views

Integrate using residue theorem

This was a question on my complex analysis take home final. Since the semester is over and grades have been posted I believe I can post it now. Let $a > 0$ and $b > 0$. Verify that ...
1
vote
1answer
35 views

complex analysis fundental theorem of caculus

Can anyone please explain how $$\int \frac{1}{(z-2)^3}dz $$ evaluated about the closed continuous path $$1+3e^{i2t\pi}$$ is 0 by the fundamental theorem of calculus?
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vote
1answer
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Evaluate Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$

I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma\,x^{-1}}\right)^s{\Gamma(\beta_1-1+s)\over \Gamma(\beta_1+\beta_2-1+s)}\,ds$$ whereby $\beta_1$, ...
2
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1answer
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Contour integral $\int_c {(z - {i^2})dz} $ over the line segment from $0$ to $1+2i$ [closed]

Hello can someone help me to solve this problem? Evaluate the integral where $c$ is the straight line segment joining $0$ and $1+2i$. $$\int\limits_c {(z - {i^2})dz} $$
0
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1answer
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Cauchy's Residue Theorem for Integral $\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$

This is a similar problem to the one I posted here. I am at this point of integration where: $$\int_{c\ -\ j\infty}^{c\ +\ j\infty} \left({\sigma \over x}\right)^s{{1-\beta^{s+1}}\over s(s+1)}\,ds$$ ...
4
votes
1answer
205 views

Evaluate Complex Integral with $\frac{\Gamma(\frac{s}{2})} {\Gamma\big({\beta +1\over 2} - {s\over 2}\big)}$

I am proving this integral: $$ \int_{c\ -\ j\infty}^{c\ +\ j\infty} \left(\,x^{-1}\sigma\beta^{1 \over 2}\,\right)^{s}\ \Gamma\left(\,s \over 2\,\right) \Gamma\left(\,{\beta +1 \over 2} - {s \over ...
2
votes
1answer
86 views

Cauchy's Residue Theorem with Multiple Gamma Functions

I previously posted a similar problem here and here. This time however I am dealing with multiple gamma functions. This is the problem I am dealing with right now: $$ \int_{c\ -\ j\infty}^{c\ +\ ...
0
votes
1answer
29 views

Is the integration of the arc in contour integration always zero?

Is the integration of the arc in contour integration always zero or is it just a most common coincidence? By arc I mean the arc $|z|=R$ and $\Im(z)\ge0$, and by integration I mean the contour ...
4
votes
2answers
101 views

Methods of evaluating $\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$

Methods of evaluating $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}$$ Firstly i know that directly: $$\int_0^{\infty}\frac{{\rm d}x}{x^2+1}=\arctan x\Bigg|_{0}^{\infty}=\frac{\pi}2$$ Also we can use the ...
1
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1answer
81 views

An integral with the $\Gamma$ function: $\int_{c- i\infty}^{c+i\infty} u^{s}\:\Gamma(\beta +s-1) \:ds$

I previously posted a similar problem here and I have solved many of the problems from the answers given with explanations. This time however I am at this point of integration where: $$\int_{c\ -\ ...
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vote
2answers
75 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 ...
1
vote
1answer
123 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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vote
1answer
75 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: ...
2
votes
1answer
62 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
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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. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
0
votes
1answer
49 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 ...
2
votes
1answer
63 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
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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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0answers
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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 ...
3
votes
2answers
90 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 ...
1
vote
1answer
72 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 ...
0
votes
0answers
65 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.
2
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1answer
54 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
45 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
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how to calculate the following integral$\int_{-\infty}^{\infty}\frac{1}{\left(t^2+\pi^2\right)^2 \cosh(t)}dt$ [closed]

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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vote
1answer
73 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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votes
3answers
163 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
123 views

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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votes
2answers
112 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 ...
3
votes
1answer
85 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} ...
0
votes
1answer
38 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 ...
2
votes
1answer
46 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 ...
4
votes
1answer
156 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 ...
8
votes
1answer
64 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, ...
1
vote
2answers
57 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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vote
3answers
86 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: $$\int^\pi_{−\pi} \frac 1 {1+\sin^2(\theta)} d\theta$$