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

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2answers
41 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 ...
9
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6answers
203 views

Evaluate $\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx$ (solution verification)

I tried to find the integral $$I=\int_0^{\infty} \frac{\log x }{(x-1)\sqrt{x}}dx \tag1$$ I substituted $x=t^2, 2tdt=dx$ and chose $\log x$ and $\sqrt{x}$ to be principal values. We have ...
1
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1answer
32 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, ...
2
votes
1answer
52 views

compute the following integral using Cauchy Integral Formula

Prove that $\int_{0}^{\pi}{e^{k\cos t}\cos (k\sin t)}=\pi$. Using Cauchy Integral Formula. But I don't know how. I want to rewrite the integral as a line integral first.
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2answers
121 views

contour integration (branch cut)

Show that $$I=\int_{-1}^1\frac{1}{(1-x^2)^{1/2}(x^2+1)}dx=\frac{\pi}{\sqrt2}$$ using contour integration, where $\;(1-x^2)^{1/2}\;$ is defined to be positive for $\; -1<x<1$.
4
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1answer
235 views

Integrate: $\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$

How to integrate using Residue theorem. $$\int_0^1 \frac{1}{\sqrt[3]{x^2 - x^3}}dx$$ How do I choose my branch-cut particularly? I was reading this article on wikiepdia and I think it is related. ...
2
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0answers
20 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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4answers
182 views

How to prove $\int^{\pi/2}_0 \log{\cos{x}} \, \mathrm{d}x = \pi/2 \log{1/2}$

ALREADY ANSWERED I was trying to prove the result that the OP of this question is given as a hint. That is to say: imagine that you are not given the hint and you need to evaluate: $$I = ...
10
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4answers
160 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} ...
4
votes
1answer
25 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 ...
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1answer
28 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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0answers
116 views

The inverse laplace transform of $p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$ can be written in Fresnel integrals?

I used the Residue theorem to solve this problem. But, I could not obtain the solution given by $$\mathscr{L}^{-1}\left( { p^{-3/2}e^{-\sqrt{pa}}\over{2\sqrt{2}}} [\cos(\sqrt{ap})+\sin(\sqrt{ap})] ...
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2answers
153 views

Evaluate $\oint_C |z|^2 dz$ around the square with vertices at $(0,0), (1,0), (1,1), (0,1)$

I don't think I quite understand how to go about this. My solution so far: $\oint_C |z|^2 dz = \oint_C (x^2 + y^2)dz = \oint_C (x^2 + y^2) d(x+iy) = \oint_C x^2 + y^2 dx + i\oint_Cx^2+y^2dy$. ...
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1answer
215 views

Problems in interpreting an integral that should be solved with residue method

Usually, when I solve an integral using residue method, I find real functions as integrands. I am not able to provide an interpretation for the following complex integral $$ \int_{-\infty}^{\infty} ...
1
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1answer
45 views

Calculation of an integral via residue.

$$\int_{-\infty}^{\infty}{{\rm d}x \over 1 + x^{2n}}$$ How to calculate this integral? I guess I need to use residue. But I looked at its solution. But it seems too complicated to me. Thus, I asked ...
2
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1answer
88 views

Evaluating $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ using complex analysis

Again, improper integrals involving $\ln(1+x^2)$ I am trying to get a result for the integral $I_{\alpha}=\int_{0}^{\infty} \frac{\ln(1+x^2)}{x^\alpha}dx$ - asked above link- using some complex ...
9
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4answers
328 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 ...
3
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1answer
372 views

Integration of Complex Logarithm

I want to prove that $$\int_0^{2\pi}\log|1-ae^{i\theta}|\, d\theta=0$$ for all $|a|\leq 1$. I can prove it easily for $|a|<1$ via power series expansion for $\log|1+(-a)e^{i\theta}|$ and then ...
5
votes
3answers
579 views

Evaluate $\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$

Evaluate by complex methods $$\int_0^{\pi} \frac{\sin^2 \theta}{(1-2a\cos\theta+a^2)(1-2b\cos\theta+b^2)}\mathrm{d\theta}, \space 0<a<b<1$$ Sis.
3
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3answers
109 views

Complex Integration : $\int_1^{1+i}\frac{1}{1+z^2}dz$

Integrate alonf the line segment from $z=1$ to $z=1+i$ : $$\int_1^{1+i}\frac{1}{1+z^2}dz$$ If I integrate, it is just the identity $tan^{-1}z$, but the answer to this question is ...
5
votes
0answers
37 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 ...
9
votes
3answers
164 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 ...
4
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1answer
177 views

Residue of $p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi x)}dx=\text{sec}1$

Show that $$p.v.\int_{-\infty}^{\infty}\frac{e^{2x}}{\cosh(\pi x)}dx=\text{sec}1$$ by integrating $\frac{e^{2z}}{\cosh(\pi z)}$ around rectangles with vertices at $z=\pm p,p+i,-p+i.$ I asked ...
3
votes
2answers
154 views

How to integrate $\int\limits_{|z| = R} \frac{|dz|}{|z-a|^2}$

I need to integrate, $\int\limits_{|z| = R} \frac{|dz|}{|z-a|^2}$ where $a$ is a complex number such that $|a|\ne R$. So first I tried polar coordinates, which gives something I cannot continue. ...
1
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2answers
91 views

Integrate: $\oint_c (x^2 + iy^2)ds$

How do I integrate the following with $|z| = 2$ and $s$ is the arc length? The answer is $8\pi(1+i)$ but I can't seem to get it. $$\oint_c (x^2 + iy^2)ds$$
2
votes
0answers
56 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 ...
3
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1answer
78 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 ...
5
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1answer
66 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 ...
2
votes
1answer
38 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 ...
6
votes
3answers
137 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: ...
2
votes
1answer
28 views

How to justify this complex substitution using contour integration

I tried to solve the laplace transform of $\cos(at)$ and $\sin(at)$ using Euler's formula. That is, $$\int^\infty_0e^{-(s-ia)t}dt\color{red}{=}\frac{1}{s-ia}\int^\infty_0e^{-t}dt=\frac{1}{s-ia}$$ ...
3
votes
1answer
207 views

Contour Integration - my solution for real integral is complex?

So I've had a crack at this contour integration question and have somehow managed to get a complex solution for a real integral... I've gone through my working a number of times but can't seem to find ...
2
votes
1answer
157 views

Finding residues for a Residue Theorem question.

The original question asks to use the residue theorem to find: $\int_0^\pi \frac{64d\theta}{(5+3cos\theta)^2} $ And by replacing $\theta$ with z's and cleaning up, I get $\frac1i\int_0^\pi \frac{246z ...
10
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1answer
101 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, ...
5
votes
2answers
98 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
52 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
votes
1answer
43 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 ...
6
votes
2answers
624 views

Evaluating $\int\limits_0^\infty \frac{\log x} {(1+x^2)^2} dx$ with residue theory

I need a little help with this question, please! I have to evaluate the real convergent improper integrals using RESIDUE THEORY (vital that I use this), using the following contour: ...
11
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3answers
534 views

Integrating $\int_0^\infty \frac{\log x}{(1+x)^3}\,\operatorname d\!x$ using residues

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,\operatorname d\!x.$$My first attempt involved trying to take a circular contour with the branch cut being the positive ...
5
votes
1answer
127 views

Integrate: $\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz$

Q. Show that : $$\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz = \frac{\sin pt}{p}$$ I considered the following contour $$\int_\Gamma \frac{e^{tz}}{z^2 + p^2}dz + \int_{a - ...
4
votes
3answers
85 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 ...
2
votes
1answer
107 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
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1answer
36 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 ...
11
votes
3answers
204 views

Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$

This question inspired me to ask the following. Prove that $$I_n = \int_0^\infty \frac{\ln x}{x^n-1}\,dx = \left(\frac{\pi}{n\sin\left(\frac{\pi}{n}\right)}\right)^2,$$ for $\Re(n)>1$. For some ...
1
vote
1answer
35 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 $J_n(x) = \frac{1}{2\pi}\int_0^{2\pi}\cos(n\theta-x\sin\theta)d\theta.$ By using contour integration with ...
6
votes
1answer
630 views

contour integral with singularity on the contour

I want to compute the following integral $$\oint_{|z|=1}\frac{\exp \left (\frac{1}{z} \right)}{z^2-1}\,dz$$ The integrand has essential singularity at the origin, and $2$-poles at $\pm 1$,which lie ...
2
votes
3answers
192 views

How to evaluate the following integral involving hyperbolic functions?

I've been thinking about using contour integration, but I can't seem to find the right combination of function and contour. Thanks for your attention. :) $$ \int_{0}^{\infty} \frac {\sinh ax \sinh ...
3
votes
0answers
60 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
votes
1answer
81 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 ...
9
votes
4answers
414 views

Differentiation wrt parameter $\int_0^\infty \sin^2(x)\cdot(x^2(x^2+1))^{-1}dx$

Use differentiation with respect to parameter obtaining a differential equation to solve $$ \int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)}dx $$ No complex variables, only this approach. Interesting ...