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

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1answer
45 views

contour integral in a region where the function doesn't have any poles

What is the value of the following contour integral? The contour is a circle with radius $0.5$ around $z=i$ point: $|z-i|<\frac{1}{2}$ $$\oint_C\frac{dz}{2-\sin z}$$ I think it is $0$ because ...
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3answers
58 views

How To Find The Length Of An Irregular Arc

How would you find out the length of an irregular arc. e.g. An arc with a base length of $10$cm and a height of $5$cm - what would be the length of that arc? Is there a specific formula I could use?
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1answer
57 views

How can I calculate the integral $ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $ [closed]

How can I calculate the integral? $$ \int_{\left| z \right| = r} \frac{dz}{(z-a)^n(z-b)^n} $$ for $ \left| a \right| < r < \left| b \right|$ and $ m, n > 1$ I tried to use the cauchy ...
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3answers
146 views

Evaluation of integral $\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$

I'm trying to evaluate the following integral: $$\mathcal{J}=\int_{0}^{\infty}\frac{\sin x}{x\left ( 1+x^2 \right )^2}\,{\rm d}x$$ Well there are $3$ poles , one lying on the real line the other on ...
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1answer
76 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
2
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2answers
113 views

Principal value of Fourier Integral

I have tried to find the principal value of $$\int_{-\infty}^\infty {\sin(2x)\over x^3}\,dx.$$ As $ {\sin(2x)\over x^3}$ is an even function, its integral may not be zero in the given limits. I ...
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1answer
49 views

Proof some 2 D Fourier transforms

Here are several Fourier transforms I used, I would like to prove those identity. I took some times to figure out how they are derived, I tried the residue theorem and other methods, but I failed, ...
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0answers
41 views

Did I apply correctly the Lebesgue dominated convergence theorem?

Let's concentrate on $$\int_0^\pi e^{iRe^{i\theta}} i d\theta$$ If $R \to \infty$, this integrand converges pointwise to $0$; plus, the modulus of the function is $= e^{-R\sin\theta} \le ...
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0answers
46 views

How to compute the covariance matrix of a random variable uniformly distributed in an ellipsoid

Suppose that x is a random variable uniformly distributed in an ellipsoid \begin{equation} x^{T}Mx\leq\delta, \end{equation} where $x\in \mathbb{R}^{n}$. Clearly, the mean of $x$ is zero. The ...
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4answers
211 views

How to compute $\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$

Could you explain to me, with details, how to compute this integral, find its principal value? $$\int_0^{\infty} \frac{\sqrt{x}}{x^2-1}\mathrm dx$$ $f(z) =\frac{\sqrt{z}}{z^2-1} = \frac{z}{z^{1/2} ...
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1answer
38 views

Integral principal value with $\cos$ and $x^2$

Could you tell me how to solve this integral? $$\int_0^{\infty} \frac{\cos x -1}{x^2}dx$$ I think I should focus on this integral $$\int_{\Gamma} \frac{e^{iz}-1}{z^2+ \varepsilon^2}$$ where ...
2
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1answer
43 views

Integral with denominator raised to n-th power, residues

I don't know how to calculate this integral: $$\int_{-\infty}^{\infty} \frac{d x}{(1+x^2)^{n+1}}$$ If we denote by $\Gamma$ a curve = semicircle centered at $0$ with radius $R$ + segment $[\ R, R]$, ...
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2answers
145 views

Pole on a contour. Problem with integration

I have a problem with calculation of the complex integral $$\int_{|z|=1}\frac{z^2+3z+2i}{(z+4)(z-1)}dz$$ Apparently integrand has a pole in $1$ lying on our circle. What can I do? I cant use Cauchy ...
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0answers
61 views

Integral with contours

I want to evaluate the integral $\displaystyle \int_0^\infty \dfrac{\ln x}{e^x+1}\,{\rm d} x$ using contour integration. At first I though using a rectangular. Problem is that I cannot establish the ...
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1answer
107 views

How does a simple elliptic integral solve this monster?

During some electromagnetics calculation regarding a loop antenna I stumbled across the following integral $$\int_0^{\pi/2} \frac{d\phi}{\big(1+\frac{k}{k-2}\cos(2\phi)\big)^{3/2}}$$ and Mathematica ...
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1answer
64 views

Complex integral problem - Two different answers! - $\oint_C\frac{dz}{z(2z+1)}$

This is from Arfken, problem #11.4.8 (7th Edition). I have to compute the complex integral, $$\oint_C\frac{dz}{z(2z+1)}$$ over the unit circle. So I took my $f(z)=\frac1{2z+1}$, and my $z_0=0$, and ...
5
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1answer
79 views

How to perform this contour integration with $\log$ in the denominator?

Let $k > 0$ and $ a>1$ be constants. As far as I can tell, the integral $$ J = \int_{-\infty}^\infty dx\frac{e^{i k x}}{1+x^2}\frac{1}{\log(a - ix)} $$ converges, since the argument of the ...
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3answers
181 views

How to prove $\int_0^1 \frac1{1+x^2}\arctan\sqrt{\frac{1-x^2}2}d x=\pi^2/24$?

Since I'm stuck at this final step of the solution here. I wished to try contour integral, taking the contour a quadrant with centre ($0$) and two finite end points of arc at $(1),(i)$: Then: ...
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1answer
120 views

Integrate $\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$ using residues

I want to find the integral $$I=\int_0^\infty \frac{dx}{(x^2+2x+12)^2}$$ using contour integration; I am familiar with the trigonometric substitution in real analysis. There are no branch cuts, ...
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2answers
134 views

How to evaluate $\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi }{48}\ln 2$?

What is the best way to calculate the following sum?$$S=\sum _{n=1}^{\infty } \frac{(-1)^{n+1} H_{2 n}^{(2)}}{n} = 2\zeta(3) - \frac \pi 2 G- \frac {\pi^2}{48}\ln 2$$ I tried putting $$f(z) = ...
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1answer
70 views

What's the difference betwen parameterizations and variable substitution for solving integrals?

Asumming I have the following integral to solve in the complex plane: $$\int \frac{dz}{z+1} $$ while $|z|=5$ which means a contour of radius 5 around zero. Is it possible to solve this integral using: ...
2
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1answer
218 views

Integration of $\ln $ around a keyhole contour

I want to evaluate the following integral: $$\int_{0}^{\infty}\frac{\ln^2 x}{x^2-x+1}{\rm d}x$$ I use the following contour in order to integrate. I considered the function $\displaystyle ...
9
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4answers
356 views

Evaluate the integral $\int_0^\infty \frac{x (\ln(x))^2}{x^4 + x^2 + 1}\text{ d}x$

What is the value of $\displaystyle\int_0^\infty \frac{x (\ln(x))^2}{x^4 + x^2 + 1}\text{ d}x$? This is a question I came up with myself. It is not homework. I constructed this example to make the ...
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1answer
94 views

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods?

Can Cauchy principal values of functions with nonsimple poles be evaluated using complex contour integration methods? In all of the examples I have seen, poles are simple and this helps to avoid ...
4
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3answers
122 views

$\int_0^\infty \frac{\log(1+x^2)}{x^2} dx $ using contour integration

I am trying to evaluate $$\int_0^\infty \frac{\log(1+x^2)}{x^2} dx $$ by using contour integration. It is possible to compute this integral using real techniques; integration by parts yields the ...
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0answers
68 views

Can this integral similar to the Fourier transformation of $\delta$ function be calculated analytically?

I want to calculate the following integral: $$\int_{-\infty}^{+\infty}dk\ \exp\left[i\big(kx-\sqrt{k(k-b)}\big)\right]$$ where $x$ and $b$ are both real. If $b=0$, the integral reduces to the Fourier ...
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0answers
76 views

Evaluating $\int_0 ^\infty \frac{\sqrt{x}}{e^x-1}dx$

I was trying to compute: $$ I_{1/2}=\int_0 ^\infty \frac{\sqrt{x}}{e^x-1}dx. $$ I know it can be recast as follows $$ I_{\alpha}=\int_0^\infty \frac{x^\alpha}{e^x-1}\ dx= \int_0^\infty ...
4
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1answer
105 views

Complex integration $\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$

I'm trying to evaluate the integral $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt$$ using complex numbers. Meaning, instead of calculating $$\int_{-\pi}^{\pi} \frac{\sin^2 t}{3+\cos t}dt,$$ I want ...
2
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0answers
69 views

Calculate contour integral (Cauchy integral formula)

I have to calculate (without refering to residue theorem) $$\int_{\partial B(2,3)} \frac{dz}{z^4-16}$$ My attempt: First, I need to find singularities of $f(z)=\frac{1}{z^4-16}$. ...
10
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3answers
213 views

Evaluating $\int_{-\infty}^{\infty}\frac{\sin ax-a \sin x}{x^3(x^2+1)} \ dx$ using contour integration

How would you compute the integral $$\int_{-\infty}^\infty \frac{\sin ax-a\sin x}{x^3(x^2+1)} \ dx ?$$ We will integrate along two circular contours and a striaghtline section between them.(Half donut ...
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1answer
32 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} ...
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2answers
91 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
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2answers
89 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
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1answer
59 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 $0\le t\le1$. ...
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2answers
31 views

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
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1answer
95 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 ...
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2answers
53 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} ...
3
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1answer
120 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 = ...
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1answer
66 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
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1answer
150 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
85 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
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1answer
76 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 ...
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1answer
38 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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1answer
86 views

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
85 views

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 $1+2i$. $$\int\limits_c {(z - {i^2})dz} $$
1
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1answer
101 views

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
209 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
101 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
31 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
111 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 ...