0
votes
0answers
57 views

Residue with half order pole?

I'm having issues evaluating the following integral using Cauchy's residue theorem. $$\int_{-\infty}^{\infty} \frac{e^{ix}}{\sqrt{x^2 - 1}} dx $$ Here's what I have tried. We have to make a ...
3
votes
0answers
73 views

Residue Integral: $\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x$

Inspired by some of the greats on this site, I've been trying to improve my residue theorem skills. I've come across the integral $$\int_0^\infty \frac{x^n - 2x + 1}{x^{2n} - 1} \mathrm{d}x,$$ where ...
0
votes
1answer
72 views

Solving contour integral

I have the integral $$ I = \int_{-\infty}^\infty dk \frac{k}{\sqrt{k^2+m^2}} e^{ikx} $$ I want to show that it decays exponentially with large $x$. I was able to solve the integral with a few tricks. ...
2
votes
1answer
108 views

using residue for integration

Hi how do u calculate the integral which have square root ? for example for this integral (because of branches points I always baffle) : $$\int_0^1 \frac{(1-x)^{1/4}\, x^{3/4}}{5-x}\, dx$$
3
votes
2answers
99 views

Evaluate by contour integration $\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$

Evaluate by contour integration [i am learning complex analysis - calculus of residues] $$\int_0^1\frac{dx}{(x^2-x^3)^{1/3}}$$ I tried by taking $x^3$ out from the denominator but that didnt work.
2
votes
0answers
55 views

inverse laplace transform of $$p^{-3/2}e^{-\sqrt{pa}}(\cos(\sqrt{ap})+\sin(\sqrt{ap}))$$

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})] ...
2
votes
3answers
75 views

Quick question on infinite complex integral

I'm supposed to find: $$ \int_0^{\infty} \frac{(\ln x)^2}{1+x^2} dx $$ I start of by finding: $$ I_1 = \oint \frac{(\ln z)^2}{1+z^2} dz $$ $$ = \oint \frac{(\ln z)^2}{(z+i)(z-i)} dz $$ Now I ...
2
votes
2answers
70 views

help with integral using residue theorem

I'm solving the following integral: $$I=\int_{-\infty}^{+\infty}\frac{x\cdot\cos x}{x^2-2x+10}\mathrm{d}x$$ I used the residue theorem as follows: ...
0
votes
0answers
56 views

Computing the contour integral of $\frac{\log(z)}{z^2 +a^2}$.

I'm still a bit insecure when it comes to complex analysis and I wondered if you guys could take a look at my solution to this problem. Let $a > 0 $ and define $$f(z) = \frac{\log(z)}{z^2 +a^2}$$ ...
4
votes
3answers
74 views

Integral $I(a,b)= P\int_{0}^{\pi}\frac{d\theta}{a-b\cos\theta}$

Hi I am trying to calculate this integral $$ I(a,b)= P\int_{0}^{\pi}\frac{d\theta}{a-b\cos\theta},\quad 0 <a<b,\quad a,b\in \mathbb{R}. $$ We can first write $$ I(a,b)=\frac{1}{2} ...
4
votes
4answers
162 views

Integral $P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx$

I am trying to calculate the following principle value integral \begin{equation} P\int_0^\infty \frac{x^{\lambda-1}}{1-x} dx \end{equation} for $\lambda \in [0,1].$ I tried to turn this into a ...
1
vote
1answer
52 views

Integrating around pie-slice domain

We are asked to show $$\int_0^{\infty}\frac{\log(x)}{x^3+1}dx=-\frac{2\pi^2}{27}$$, and $$\int_0^{\infty}\frac{1}{x^3+1}dx=\frac{2\pi}{3\sqrt{3}}$$ By integrating around a pie slice with angle ...
1
vote
1answer
72 views

Calculate integral using the residue theorem

I want to calculate the integral $$\int_C {{z^2-2z}\over{(z+1)^2(z^2+4)}}dz$$, where $C=\{z:|z|=4\}$ I want to use the Residue theorem to tackle this integral. Now, $f(z)$ has a pole of degree 2 at ...
0
votes
1answer
70 views

Application of Residue Theorem and limits

I am trying the following problem from Fisher's Complex Variables book: If $f$ is analytic on a plane except at poles $\gamma_1, \cdots \gamma_N$ and none of them are integers and ...
2
votes
1answer
44 views

I want to compute $\int_0^\infty \frac{x^t}{1+x^2}dx \; \forall t \in (-1,1)$ using residue theroem.

I want to compute $$\int_0^\infty \frac{x^t}{1+x^2}dx \qquad \forall t \in (-1,1)$$ using residue theroem. I consider $$f(z) = \frac{z^t}{1+z^2}$$ I find two pole of order 1 in $z=i$ and $z=-i$ with ...
2
votes
2answers
66 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
0answers
20 views

computing integral using residue theory [duplicate]

I want to compute the integral $\int_{-\infty}^\infty {x^4\over {1+x^8}}dx$ by using residue theory. I find the zero of $Q(x)$ is $i^{1/4}$. Do I have to factor the denominator into 8 different ...
2
votes
2answers
63 views

Find $\int_\Gamma\frac{2z+j}{z^3(z^2+1)}\mathrm{d}z$ where $Γ:|z-1-i| = 2$

pls, some ideas for integral solution (residue theory)? $$\int_\Gamma\dfrac{2z+j}{z^3(z^2+1)}\mathrm{d}z$$ Where $Γ:|z-1-i| = 2$ is positively oriented circle. Thx, for help!
3
votes
1answer
133 views

prove $\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $

prove $$\int_{0}^{2\pi}(1-\cos x)^n\cos nx dx=(-1)^n\frac{\pi}{2^{n-1}} $$ I tried with $2\cos x =z+\frac1z$ then use residue theorem but I faced some troubles my try is : $2\cos x =z+\frac1z$ ...
3
votes
2answers
168 views

Integral $\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$

Consider $$\int_{0}^{2\pi}\log|e^{i \theta}-1|d \theta$$ Is it equal to $0$ ? Why ? Any hint ?
1
vote
3answers
76 views

Integral $\int_{0}^{\infty}e^{-ax}\cos (bx)\operatorname d\!x$

I want to evaluate the following integral via complex analysis $$\int\limits_{x=0}^{x=\infty}e^{-ax}\cos (bx)\operatorname d\!x \ \ ,\ \ a >0$$ Which function/ contour should I consider ?
3
votes
2answers
61 views

Using residue to find a complex integral

Given the following: $$\int_{\varGamma_R} {z\,dz\over e^{2\pi iz^2}-1}, \ \ \ \varGamma_R=\{z\in \Bbb C:|z|=R\},\quad n<R^2<n+1,\,n\in\Bbb N.$$ I want to use the residue for this, but I can't ...
4
votes
1answer
155 views

Integral Using Harmonic Functions

Evaluate the integral: $$\int^{2 \pi}_0 \dfrac{\cos^2 \theta}{|2e^{i\theta}-z|^2} \, d \theta \qquad \mbox {when} \, |z| \neq 2.$$ Now, I thought about trying to change this to look like a Poisson ...
2
votes
2answers
93 views

About the integral $\int_{0}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}$

I need to prove the following identity: $$\forall a>0,\qquad\int \limits_{-\infty}^{+\infty}\frac{\sin(ax)\,dx}{x(x^2+1)}=\pi(1-e^{-a}).$$ I think it can be proven using Laurent series. I tried ...
0
votes
4answers
58 views

Integrating $t^{2r-1} / t^{2k} (1+t^2)^{r+1}$

Let $k$ and $r$ be natural numbers such that $1 \leq k \leq r$. I want to calculate $$ \int_0^\infty \frac{t^{2r-1}}{t^{2k}(1+t^2)^{r+1}} dt. $$ Since the integrand is an odd function the standard ...
4
votes
4answers
292 views

Calculating $\int_0^\infty \frac {\sin^2x}{x^2}dx$ using the Residue Theorem.

I am trying to compute the following integral using the Residue Theorem but am quite stuck: $$\int_0^\infty \frac{\sin^2x}{x^2}dx$$ I have tried applying Jordan's lemma, having written $\sin(x)$ as ...
0
votes
1answer
50 views

Integral $\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$

I want to solve the integral $$\int_{0}^{+\infty}\frac{t \sin(t)}{t^{2}+b^{2}}dt$$ Which function and contour should I consider ?
1
vote
2answers
144 views

$\int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$ with residue calculus

I'm trying to compute $\displaystyle \int_{-\infty}^\infty \frac{e^{ax}}{1+e^x}dx$, $(0<a<1)$ Let $f$ denote the integrand. I'm using the rectangular contour given by the following curves: ...
16
votes
2answers
421 views

Show $\int_0^{\pi/3} \big((\sqrt{3}\cos x-\sin x)\sin x\big)^{1/2}\cos x \,dx =\frac{\pi\sqrt{3}}{8\sqrt{2}}. $

I have run a FORTRAN code and I have obtained strong evidence that $$\int_0^{\pi/3} \!\! \big((\sqrt{3}\cos\vartheta-\sin\vartheta)\sin\vartheta\big)^{\!1/2}\!\cos\vartheta \,d\vartheta ...
3
votes
1answer
138 views

Would like help with a contour integral.

Disclaimer: the knowledge I have about contour integration is solely from the book "Mathematical Methods in the Physical Sciences" by Mary L. Boas. I am trying to understand how the following ...
3
votes
2answers
157 views

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem.

How to compute the integral $\int_0^\infty\frac{x}{e^x+1}dx$ using the Residue theorem, just as the title says. I have used rectangles, circles to do, but without any progress. By changing variable ...
2
votes
2answers
51 views

Problem with Mellin Barnes type integral

Using the Mellin Barnes technique for a certain Feynman integral, I arrive at $$ I= \frac1{2\pi i} \int\limits_{-i\infty}^{i\infty} dz\; \Gamma^4\left(\frac12 + z\right) ...
1
vote
1answer
148 views

Evaluating real improper integral by residues

I've been trying to solve this integral and have been getting nowhere: $$ \int_0^\infty \frac{dx}{(1+x^2)x^a} \;,\; 0<a<1 $$ The solution says that $$ \int_0^\infty \frac{dx}{(1+x^2)x^a} = ...
11
votes
3answers
426 views

Calculate $\displaystyle \int_0^\infty \frac{\ln x}{1 + x^4} \mathrm{d}x$ using residue calculus

I need to evaluate this integral using calculus of residues: $$\int_0^\infty\frac{\ln(x)}{1+x^4}\mathrm{d}x$$ I know I need to consider $\displaystyle ...
4
votes
2answers
331 views

Computing $\int_{-\infty}^\infty \frac{\sin x}{x} \mathrm{d}x$ with residue calculus

This refers back to the integral of $\frac{\sin(x)}x = \frac\pi2$ already posted. How do I arrive at $\frac\pi2$ using the residue theorem? I'm at the following point: $$\int \frac{e^{iz}}{z} - \int ...
2
votes
2answers
230 views

What are the reasons for using a semi-circle in upper half plane of $\mathbb{C}$ for contour integration?

Why is it that when one in considering contour integration of a real function, such as $$ \int_{-\infty}^{\infty} \frac{dx}{1+x^2}$$ the contour in the complex plane used is the following: ...
5
votes
3answers
184 views

Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Evaluate the integral $$\int_{-\infty}^{\infty} \frac{1}{(1 + x^2)^{n+1}} dx. $$ I know that it equals $2\pi i$(the sum of the residues; at $z_k$) where $z_k$ are the poles of the function. I ...
7
votes
3answers
369 views

Evaluate $\int_{0}^{\infty}\dfrac{\mathrm dx}{(e^{\pi x}+e^{-\pi x})(16+x^2)}$

Find the integral $$I=\int_{0}^{\infty}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ My try:let $x=-t$ $$I=\int_{-\infty}^{0}\dfrac{1}{(e^{\pi x}+e^{-\pi x})(16+x^2)}dx$$ so ...
3
votes
2answers
197 views

Evaluating $\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$

I have been asked to evaluate $$\int_{-\infty}^\infty \frac{dx}{\cosh(x-a)\cos(2x)}$$. I'm deliberating on whether this indefinite integral exists or not. The integrand diverges when ...
1
vote
1answer
198 views

Can the direction of Contour Integral be affect to the result of integration?

Now i doing the home work about Residue Integration and i doubt that "Can the direction of Contour Integral be affect to the result of integration?" I mean with the same shape of contour but direction ...
4
votes
2answers
304 views

Why do we use only upper half plane to do Residue Integration?

In the class of mathematics my professor showed me that "how to use the residue integration method?". And i doubt at almost the last step that professor did. he made a contour over the upper half ...
1
vote
3answers
74 views

Evaluation of the integral.

$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$ The integration is taken along the imaginary axis, an integer ...
2
votes
2answers
110 views

Residue theorem, $\int_{-\infty}^{\infty} e^{-ikx}(1-ika^2)^{-m} dk$ with integer $m$

I am trying to solve this integral $\int e^{-ikx}(1-ika^2)^{-m} dk$ using the residue theorem, but I cannot find the residue of the function. $$\frac{1}{(1-ika^2)^{-m}}=\sum (ika^2)^n(-1)^n ...
4
votes
1answer
185 views

Expressing an integral in terms of the Bernoulli numbers

In Ahlfors' Complex Analysis text, the Bernoulli numbers, $B_k$, are defined as the coefficients in a Laurent development: $$(e^z-1)^{-1}=\frac{1}{z}-\frac{1}{2}+ \sum_1^\infty (-1)^{k-1} ...
2
votes
2answers
87 views
10
votes
3answers
420 views

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

I am trying to use residues to compute $$\int_0^\infty\frac{\log x}{(1+x)^3}\,dx.$$My first attempt involved trying to take a circular contour with the branch cut being the positive real axis, but ...
2
votes
2answers
125 views

Finding a definite integral by residue integration?

I have a probability distribution of the form $$\frac{k_1}{(k_2 x^2+k_3 x+k_4)^n}$$ and I want to find the mean and variance—but am running into problems with that. The mean would be ...
3
votes
0answers
178 views

Turning a Line Integral into a Contour one

I'm trying to compute an integral appearing in the article "On Determinants of Laplacians on Riemann Surfaces" of D'Hoker and Phong (page 541). It is as following. Fix $B\in \mathbb{R}_+$ and let ...
8
votes
7answers
441 views

How to calculate $ \int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4} $?

I would like to calculate $$\int_{0}^{\infty} \frac{ x^2 \log(x) }{1 + x^4}$$ by means of the Residue Theorem. This is what I tried so far: We can define a path $\alpha$ that consists of half a ...
4
votes
1answer
89 views

Hint to compute the following integral

Can someone give a hint on how to solve the following integral? $$ \int_{0}^{2N\pi} \frac{-R\left(\xi t - r\right)\cos\left(t\right) + \xi R\sin\left(t\right)} {\left[R^{2} + \left(\xi t - ...