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

learn more… | top users | synonyms

2
votes
2answers
88 views

Evaluating $\int_c\frac{1}{\sin\frac{1}{z}}\text{d}z$ over $C= \{z\big\vert|z|=\frac{1}{5}\}$

Evaluate $$\int\limits_{|z|=\frac{1}{5}} \frac{1}{\sin\frac{1}{z}}\text{d}z$$ My attempt: I know that this function has non isolated singularity at $0$, and simple poles at $\frac{1}{n \pi}$. ...
6
votes
1answer
118 views

Prove that a complex-valued entire function is identically zero.

Suppose $f$ is entire and $$\iint_\mathbb{C}|f(z)|^2dxdy < \infty$$Prove that $f\equiv 0.$ So far I have: Suppose $f$ is bounded. Then $f$ is constant by virtue of Liouville and so the ...
0
votes
2answers
50 views

Contour Integral finding Poles

I have an integral to solve using appropriate contour integrals; The question is like this. $$\int\limits_0^{2\pi}\frac{\cos(n\theta)d\theta}{1+2p\cos(\theta)+p^2}$$ $$-1<p<1$$ So I thought ...
1
vote
1answer
35 views

Contour Integral of $1/(zcos(z))$ around the circle $\lvert z \rvert = n\pi$

This is part of a past exam question from a second year undergraduate complex variable theory course. I am attempting to show that the integral in the title tends to zero as n goes to infinity. The ...
0
votes
1answer
28 views

Complex analysis contour integration calculation check if it is right

Calculate $\int{ \frac{e^z}{z^2(z^2+3)}}dz$ over the rectangle $x=2,x=-2 ,y=2,y=-2$. What i did is find the roots of $z^2+3$ break the $\frac{1}{z^2+3}$ into $\frac{-i/6}{z-3i} +\frac{i/6}{z+3i}$ ...
3
votes
1answer
164 views

Contour integral for finding $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx$

I can't prove the following result: $\displaystyle\int_{0}^{\infty}\frac{\ln x}{(x+a)^2+b^2}dx=\frac{\ln \sqrt{a^2+b^2}}{b}\arctan\frac{b}{a}$ for all $a,b \in \mathbb{R}.$ Well, I consider $\...
0
votes
0answers
19 views

Contour integration problem (not sure if an answer exists)

Here's a tough contour integral question which I haven't been able to solve. It arose out of an amateur fiddling and may very well have a critical error that makes it unable to be evaluated. $f(z)=\...
4
votes
4answers
123 views

Evaluating $\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$ using contour integration

I need your help with this integral: $$\int_{0}^\infty \frac{\log x \, dx}{\sqrt x(x^2+a^2)^2}$$ where $a>0$. I have tried some complex integration methods, but none seems adequate for this ...
3
votes
4answers
225 views

How to find $L = \int_0^1 \frac{dx}{1+{x^8}}$

Let $L = \displaystyle \int_0^1 \frac{dx}{1+{x^8}}$ . Then $L < 1$ $L > 1$ $L < \frac{\pi}{4}$ $L > \frac{\pi}{4}$ I got some idea from this video link. But got stuck while evaluating ...
0
votes
3answers
70 views

Evaluating the real integral $\int_{0}^{2\pi}\frac{1}{2+\sin\theta}d\theta$ using complex analysis

I thought it's value would be zero, since the complex integrand: $$\Im\left(\int_{C}\frac{1}{2+e^{i\theta}}d\theta\right)$$ Where $C$ is the unit disc, is nonsingular. Also $e^{iz}\ne -2$ for any $z$...
0
votes
1answer
17 views

Contour Integration along a line segment

This may sound like a silly question, but was just wondering if someone can clear this up for me Consider the line segment joining the points a,b. Therefore we have f(t)=a+t(b-a) where t is ...
3
votes
3answers
82 views

Evaluating the integral of $\frac{\cos(x) - e^{-x}}{x}$ using contour integration

I am trying to evaluate the value of $$\int_0^\infty\frac{\cos(x) - e^{-x}}{x}dx$$. I am assuming I am supposed to use contour integration, as I was required just before to calculate the value of $$\...
3
votes
2answers
77 views

Prove that: $\frac{2\pi i}{(1 - e^{2i\pi/n})\prod_{k=0, k \neq 1}^{n-1} (e^{i\pi/n} - e^{i(2k-1)\pi/n})} = \frac{\pi/n}{\sin(\pi/n)}$

I am trying to find $\int_0^{\infty} \frac{dx}{1 + x^n}$ using contour integration. I did the computation by taking the contour $[0,R] \cup \gamma_R \cup [R e^{2i\pi/n}, 0]$, with $\gamma_R$ the arc ...
2
votes
4answers
76 views

Evaluation of the principal value of $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3} \, dx$

I'm trying to evaluate an integral $\int\limits_{-\infty}^\infty \frac{\sin 2x}{x^3}\,dx$ using Cauchy's theorem. Considering an integral from $-R$ to $-\epsilon$, then a semicircular indentation ...
0
votes
2answers
46 views

Contour integration with logarithms

I'm having trouble calculating the below integral to get the right answer: $$\frac{1}{2\pi i}\int_\gamma \frac{3}{z-2}\; dz$$ where $\gamma$ is parametrised by $\gamma(t)=3e^{it}, t\in [0,2\pi]$. So ...
0
votes
1answer
33 views

Recommendations for tutorials specifically devoted to real integration using contour integral techniques.

Complex analysis, and in particular contour integrals and the residue theory have proved a very powerful tool in computing a large class of real function integrals which would be quite troublesome to ...
1
vote
2answers
26 views

expressing contour integral in different form

Hi I have a short question regarding contour integration: Given that $f(z)$ is a continuous function over a rectifiable contour $z = x + iy$. If $f(z) = u(x,y) + iv(x,y)$, why does it follow that the ...
8
votes
3answers
155 views

Need help with $\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$

I'm having trouble trying to evaluate this definite integral. Mathematica didn't help much. $$\int_{-\infty}^\infty \frac{x^2 \, dx}{x^4+2a^2x^2+b^4}$$ where $a$, $b$ $\in \Bbb R^+$. Is it possible ...
1
vote
4answers
59 views

Computing the residue of a rational function

The real integral I am trying to compute with residues/contour integration is $\int_{-\infty}^{\infty}\frac{x^2}{(x^2+a^2)^3} \,dx$ For $a$ positive and by using the complex integral $$\int_{C_R}\...
0
votes
1answer
43 views

Calculating Infinite Real Integrals Using Residues

I want to calculate the following real integral using residues and I am unsure how to proceed. $$\int_{-\infty}^{+\infty}\frac{1- x^2}{1+ x^4} dx$$ I know I must change this to a contour integral so ...
0
votes
1answer
52 views

What is the method to use the generalised Cauchy Integral Formula

Past Paper Question: a) State the generalized form of Cauchy’s integral theorem b)Evaluate $$\displaystyle f(z)=\int_{\gamma}\frac{z^2}{\biggr(z-\dfrac{\pi}{4}\biggl)^3} dz$$ where $\gamma$ is ...
0
votes
0answers
22 views

$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$

$$\int \frac{\cos z}{z(z+2)}\mathop{\mathrm{d}z}$$ traversing the unit circle counterclockwise. So the singularities are $z=0$ and $z=-2$ but the second is outside the unit circle so it isn't ...
0
votes
3answers
54 views

$\int \frac{2+\sin(z)}{z} dz$

Please bear in mind that I am trying to teach myself complex integration having never taken a course in complex analysis, so assume I know very little. $\int \frac{2+\sin(z)}{z} dz$ traversing the ...
1
vote
1answer
31 views

$f_n$ converges uniformly on $\partial \Omega$ then $f_n$ converges uniformly on $\bar{\Omega}$

The problem states that $f_n$ is a sequence of functions which are continuous on the closure of $\Omega$ and holomorphic on $\Omega$ where $\Omega$ is a bounded region and were asked to show that if $...
1
vote
0answers
47 views

$\int_{-\infty}^\infty \frac{dz}{z - z_0}$ by contour integration

Consider the integral $\int_{-\infty}^\infty \frac{dz}{z - z_0}$. It has a simple pole at $z = z_0$. Assume $\Im (z_0) < 0$ so the pole is in lower half-plane. Divide $$ \oint_{C_0} = \int_{-R}^R +...
1
vote
1answer
29 views

Problem with integration limits using spherical substitution

Good night, i have a problem with this integral, please help me with the integration limits. \begin{align} \int_{-1}^{1}\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\int_{-\sqrt{1-x^{2}-y^{2}}}^{\sqrt{1-x^{...
0
votes
0answers
35 views

An Alternate proof of Nyquist-Shannon

This problem is from Basic Complex Analysis, Part 2A, by Barry Simon. This problem will provide an alternate proof of the strong from of the Nyquist-Shannon sampling theorem (Theorem 6.6.16 of Part 1)...
1
vote
1answer
17 views

Problem with integration limits with cylindric cordinates.

Good night, i have a problem when i go to verify the integration limit $0\leq\theta\leq\varPi/2$ because i think the integration limit go to $0\leq\theta\leq\varPi$ because is an half a circle. $\...
3
votes
1answer
74 views

Integral of $p(x)\operatorname{csch}(x)$

I'd like to calculate the following integral $$\int_{-\infty}^{+\infty}\frac{x^4 \left(\frac 1 {a^2+x^2} +\frac 1 {b^2+x^2}\right)}{\sinh^2(x\pi /c)} \, dx$$ where $a$, $b$ and $c$ are positive ...
1
vote
1answer
24 views

Integral estimation

Is it true that $$\int_{-\pi}^{\pi}\left |\frac{iR^{x}e^{i \theta x}}{1-Re^{i\theta}}\right |\,\mathrm{d}\theta \leq 2\pi R\frac{R^{x}}{1-R}\xrightarrow{R\to\infty}0$$ $$\int_{-\pi}^{\pi}\left |\frac{...
4
votes
0answers
47 views

Complex Contour Integral Involving Arg(z)

My question is regarding the following complex integral: $$\int_\gamma\frac{\operatorname{Arg}(z)}{z} dz$$ where $\gamma$ is the curve defined by:$\quad$ $\gamma(t) = e^{it}, 0\leq t\leq \frac{\pi}{...
0
votes
0answers
12 views

Splitting a contour integral into separate integrals?

Suppose you had a complex contour integral of the form $$ \int_{\alpha + \beta} f(z) \; dz $$ Is this equivalent to $$ \int_{\alpha} f(z) \; dz + \int_{\beta} f(z) \; dz $$ Thanks
0
votes
0answers
27 views

Integral problem with branch point from Physics

The question come from a Summation like this $${ \sum _{ { z=i\omega }_{ n } } { \frac { -\alpha E\pi }{ 4{ z }^{ 3 }\sqrt { -\alpha -z } } } }$$ I can use Cauchy theorem to transform it to a ...
0
votes
1answer
28 views

Proof of the Primitive Existence Theorem

In the first section of the proof of the primitive existence theorem we are trying to show that $$F\left(z+h\right)-F\left(z\right)=\int_{L_{z,z+h}}f$$ where $\alpha, z,z+h$ are collinear and $\alpha=...
2
votes
0answers
17 views

Choice of branch cut

I'm trying to rewrite an integral of the form $$\int_{-∞}^∞ (x^2+k^2)^{-s/2}e^{i(xc_1+kc_2)}dx$$ ($s>0,c_1,c_2,k\in\mathbb R$) in such a way that it is positive (for some $s$). To do so I'd like ...
1
vote
0answers
20 views

How to prove reflection positivity for $|x|^{-p}$ using Fourier transform (and contour integrals)

This question looks quite lengthy because I'm sketching the proof in the lecture - the two questions (look out for something bold) are actually relatively short. I need some help with a proof in our ...
0
votes
0answers
27 views

Evaluate the following contour integral

Evaluate the integral $$\int_{C}\left(\exp\left(z\right)-\dfrac{2}{z^3}\right)dz$$ where C is any contour from $i$ to $-i$ which does not pass through $0$ I understand that contour integration can ...
0
votes
1answer
16 views

Showing that a function is the restriction of another holomorphic function

Let $f$ be holomorphic on the annulus $\{z:\;1-\epsilon<|z|<1+\epsilon\}$. Define: $$\phi:\;D\to\mathbb{C};\quad \phi(w)=\frac{1}{2\pi i}\int_C \frac{f(z)}{z-w}\,\mathrm{d}z$$ where $C$ is the ...
2
votes
0answers
34 views

$ f(z) = \int _{0} ^\infty e^{-z t^2} dt $ is holomorphic in $Re(z) > 0$

I've been trying to find a reasonable way to approach this but nothing leads to a reasonable result. With $z = a + ib$ the last thing I tried looking at was $$| \frac{f(z) - f(z-h)}{h}| = |\frac{1}{...
0
votes
0answers
22 views

conversion from path integral to contour integral

I'm considering the following integral $$I=\int_{-i\infty-\epsilon}^{i\infty-\epsilon}\frac{\psi(v)}{v}\;dv$$ I want to use the following substitution to convert the path integral to a contour ...
4
votes
2answers
130 views

Evaluate $\int_0^{2\pi}\frac{\sin^2(x)}{a + b\cos(x)}\ dx$ using a suitable contour

I need to find a good contour for $\int_0^{2\pi}\frac{\sin^2(x)}{a + b\cos(x)}\ dx$ but I don't know which one to choose. Both a semicircular, and rectangular contour look ugly for this. I've been ...
2
votes
1answer
24 views

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it)$

Show that $\displaystyle \int_{\gamma_R}\exp(-w^2)\,\mathrm dw\to 0~(R\to\infty)$ along $\gamma_R(t)=R\exp(\mathrm it),t\in\left[0,\frac{\pi}{4}\right]$. Hint. Use that $\cos 2t\geq 1-\frac{4}{\...
2
votes
1answer
50 views

Evaluate $I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$

This is the last part of a multistage evaluation of $I(x)=\Gamma(x)\Gamma(1-x)$. Through various substitutions we get the integral $$I(x)=\int_{-\infty}^{\infty}\frac{e^{xv}}{1+e^v}dv$$ We're also ...
1
vote
0answers
11 views

Compute an integral involving exterior Riemann mapping

Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. There exists a unique exterior conformal mapping $\Phi$ from $\...
4
votes
1answer
157 views

Show that $\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta = \frac {\pi r^n}{1-r^2}$

I am trying to calculate $$I=\int_0^{\pi}\frac {\cos {n \theta}}{1-2r\cos \theta+r^2}\, \mathrm d \theta$$ where $r\in(0,1)$ I tried substituting $u = e^{2 i \theta}$ and using the Cauchy integral ...
0
votes
0answers
23 views

Double Integral with Residues

I'm trying to solve the integral $$\int_a^b\int_a^b\frac{dxdy}{1+\left(x^2+y^2\right)^\alpha}$$ where the constant $\alpha$ is real-valued and in the range $\alpha\in[1/2,\infty)$. The bounds $a$ ...
10
votes
0answers
794 views

Integral of rational function over $\mathbb{H}^4$

Suppose I have a rational function of $8$ coordinates $a,b,c,d,e,f,g,h$ that I want to integrate over $\mathbb{H}^4$: ...
0
votes
2answers
57 views

Complex integration problem via Cauchy's integral formula

I want to integrate the following :$$\int_{|z|=2} \frac{dz}{z^{2}-1}$$ in the positive direction. So my idea is two split the integral into a sum of two integral , something like $$\int_{|z|=2} \frac{...
4
votes
1answer
41 views

How can I evaluate this contour integral?

Suppose we have the following contour integral, in the complex plane: $$ \int_{\gamma} \frac{e^{\frac{1}{z}}}{z^{2}} \; dz $$ where $\gamma (t) = e^{it}$ for $0 \leq t \leq 2 \pi$. To solve this, I ...
0
votes
0answers
61 views

Find all possible values of the integral

Find all possible values of $\displaystyle I= \int_C \frac{dz}{1+z^2}$, where $C$ is a curve with initial point $0$ and final point $1$ that does not meet the poles of $\dfrac{1} {1+z^2}$. It looks ...