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

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10
votes
4answers
136 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
24 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 ...
1
vote
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 $[\ R, R]$, ...
4
votes
0answers
34 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 ...
3
votes
1answer
76 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 ...
2
votes
1answer
36 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
votes
1answer
64 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 ...
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: ...
10
votes
1answer
98 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
97 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) = ...
1
vote
1answer
51 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
42 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 ...
8
votes
4answers
325 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 ...
1
vote
1answer
34 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
votes
3answers
84 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
0answers
55 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
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 ...
1
vote
0answers
36 views

Caculate 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 $f(z)=\frac{1}{z^4-16}$. ...
9
votes
3answers
163 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 ...
2
votes
1answer
28 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
votes
2answers
71 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
83 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
50 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
25 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
votes
1answer
71 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
47 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
votes
1answer
106 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
58 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
104 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} ...
2
votes
1answer
53 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
62 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
29 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?
1
vote
1answer
76 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
votes
1answer
59 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 $1+2i$. $$\int\limits_c {(z - {i^2})dz} $$
0
votes
1answer
93 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
193 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
63 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
26 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
98 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
vote
1answer
68 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\ -\ ...
1
vote
2answers
57 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
120 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 = ...
1
vote
1answer
73 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
59 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 ...
1
vote
0answers
49 views

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 ...
1
vote
3answers
116 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. $$I = \int_{0}^{\pi} \log(\sin(x)) dx$$ ...
0
votes
1answer
42 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
61 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
2
votes
2answers
63 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 $2\pi i(2\omega^2-\omega-2)$.