Complex integral Definition and 125 Threads

I’ve attached my attempt. I’ve tried to use triangle inequality formula to attempt, but it seems I got the value which is larger than 1. Which step am I wrong? Also, it seems I cannot neglect the minus sign in front of e^(N+1/2)*2pi. How can I deal with that?

Hi, I have some question about evaluating a cosine function from ##-\infty## to ##\infty##. I saw for a cosine function evaluate from ##-\infty## to ##\infty## I can change the limits from 0 to ##\infty##. I have a idea why, but I can't convince myself, furthermore, is it always the case no...

I was trying to calculate an integral of form: $$\int_{-\infty}^\infty dx \frac{e^{iax}}{x^2}$$ using contour integration, with ##a>0## above. So I would calculate a contour integral with contour being a semicircle that goes along the real axis, closing it in positive direction in the upper...

I have a complicated function to integrate from -\infty to \infty . I = \int_{-\infty}^{\infty}\frac{(2k^2 - \Omega^2)(I_0^2(\Omega) + I_2(\Omega)^2) - \Omega^2 I_0(\Omega) I_2(\Omega)}{\sqrt{k^2 - \Omega^2}} \Omega d\Omega Where I0I0 and I2I2 are functions containing Hankel functions as...

As this function has no singularities the residue theorem cannot be applied. Can you help me a bit?

Homework Statement The following is a problem from "Applied Complex Variables for Scientists and Engineers" It states: The following integral occurs in the quantum theory of collisions: $$I=\int_{-\infty}^{\infty} \frac {sin(t)} {t}e^{ipt} \, dt$$ where p is real. Show that $$I=\begin{cases}0 &...

Homework Statement ##\int_{0}^{2\pi} cos^2(\frac{pi}{6}+2e^{i\theta})d\theta##. I am not sure if I am doing this write. Help me out. Thanks! Homework Equations Cauchy-Goursat's Theorem The Attempt at a Solution Let ##z(\theta)=2e^{i\theta}##, ##\theta \in [0,2\pi]##. Then the complex integral...

Hey, I tried to construct the derivation of the integral C with respect to Y: $$ \frac{\partial C}{\partial Y} = ? $$ $$ C = \frac{2}{\pi} \int_0^{\infty} Re(d(\alpha) \frac{exp(-i \cdot ln(f))}{i \alpha}) d \alpha $$ with $$d(\alpha) = exp(i \alpha (b + ln(Y)) - u) \cdot exp(v(\alpha) + z...

The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...

Homework Statement I have the following integral I wish to solve (preferably analytically): $$ I(x,t) = \int_{-\infty}^{0} \exp{[-(\sigma^2 + i\frac{t}{2})p^2 + (2\sigma ^2 p_a + ix)p]} \ dp$$ where ##x## ranges from ##-\infty## to ##\infty## and ##t## from ##0## to ##\infty##. ##\sigma##...

Homework Statement I have an integral $$\int_{-\infty}^{0} e^{-(jp - c)^2} \ dp$$ where j and c are complex, which I'd like to write in terms of ## \text{erf}## I'd like to know what would happen to the integral limits as I make the change of variables ##t = jp - c##. 1) As ##p## tends...

Hello! If I have a real integral between ##-\infty## and ##+\infty## and the function to be integrated is holomorphic in the whole complex plane except for a finite number of points on the real line does it matter how I make the path around the poles on the real line? I.e. if I integrate on the...

Hello! I found a proof in my physics books and at a step it says that: ##\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx \sim_{t \to \infty} e^{-imt}##. Any advice on how to prove this?

I am trying to do the following integral: $$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)}.$$ Wolfram alpha - http://www.wolframalpha.com/input/?i=integrate+(cos(x))^(1/2)+dx+from+x=pi+to+3pi gives me $$\int_{\pi f}^{3\pi f} dx \sqrt{\cos(x/f)} = 4f E(2) = 2.39628f + 2.39628if,$$ where E is the...

I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...

Homework Statement I'm having some trouble evaluating the integral $$\int^\infty_{-\infty} \frac{\sqrt{2a}}{\sqrt{\pi}}e^{-2ax^2}e^{-ikx}dx$$ Where a and k are positive constants Homework Equations I've been given the following integral results which may be of help $$\int^\infty_{-\infty}...

Hey everyone, I am trying to evaluate the following integral: \int z(z+1)cosh(1/z) dz with a C of |z| = 1. Can someone please guide me with how to start? I have tried to parametrise the integral in terms of t so that z(t) = e^it however the algebra doesn't seem to work...

Hi, I have a difficult time trying to perform the following integral, $$ j({T}, \Omega)=\int_0^{ T} d\tau \frac{\tau^2\exp(-i\Omega\tau)}{(\tau-i\epsilon)^2(\tau+i\epsilon)^2} $$ The problem is that the poles ##\pm i\epsilon## when taking the limit ##\epsilon\rightarrow 0## are located at...

An old exam question is: Evaluate $ \oint \frac{e^{iz}}{z^3}dz $ where the contour is a square of sides a, centered at 0. This has a simple pole of order 3 at z = 0 Perhaps using residues, $ Res(f,0) = \frac{1}{2!}\lim_{{z}\to{0}}\d{^2{}}{{z}^2}z^2 \frac{e^{iz}}{z^3} =...

An old exam has: Evaluate $ \oint\frac{dz}{z(2z+1)} $, where the contour is a unit circle This look good for the residue theorem, it has 2 simple poles at 0, $-\frac{1}{2}$ $ Res(f, 0)= \lim_{{z}\to{0}}z\frac{1}{z(2z+1)}=1$ $ Res(f, -\frac{1}{2})=...

Hi. If a real integral between 2 values gives the area under that curve between those 2 values what does a complex integral give between 2 values ?

This relates to z-transform causality, but I'll try to phrase it as a complex analysis question. Suppose I have a function ##X(z)## whose poles are all inside the unit circle, and which has the property \lim_{|z|\to\infty} \frac{X(z)}{z} = 0 Is that sufficient to guarantee that \frac{1}{2\pi...

Homework Statement Describe all the singularities of the function ##g(z)=\frac{z}{1-\cos{z}}## inside ##C## and calculate the integral ## \int_C \frac{z}{1-\cos{z}}dz, ## where ##C=\{z:|z|=1\}## and positively oriented. Homework Equations [/B] Residue theorem: Let C be a simple closed...

How can I solve the integral below? ## \int_{-\infty}^{\infty} \sqrt{k^2+m^2} e^{izk} dk ## I thought about contour integration but, as you can see, it doesn't satisfy Jordan's lemma. Also no substitution comes to my mind!

Let $[a,b]$ be a closed real interval. Let $f:[a,b] \to \mathbb{C}$ be a continuous complex-valued function. Then $$\bigg|\int_{b}^{a} f(t)dt \ \bigg| \leq \int_{b}^{a} \bigg|f(t)\bigg| dt,$$ where the first integral is a complex integral, and the second integral is a definite real integral...

Hello, I am trying to understand how to get the residue as given by wolfram : http://www.wolframalpha.com/input/?i=residue+of+e^{Sqrt[x^2+%2B+1]}%2F%28x^2+%2B+1%29^2 The issue I am facing is - since it is a second order pole, when I try to different e^{\sqrt{x^+1}} I get a \sqrt{x^+1}...

I'm solving two different definite integrals of functions \frac{sin(z)}{z} and \frac{cos(z)}{e^z+e^{-z}} with complex analysis and the residue theorem, and in the solutions they replace both sin(z) and cos(z) with e^{iz} why is this possible?

integrate e^z/(1-cosz) dz over circle of radius, say 2 i can't seem to recall how it is done. singularity at z=0 2*pi*i * res (at z=0) would be the solution any shortcut to find this residue?

integral This is my work but I got a very looong integral to solve after substitute tg (x/2) based in my former exercise it remains 4/ (1-z^2)(3-z^2)

Homework Statement . Let ##\gamma_r:[0,\pi] \to \mathbb C## be given by ##\gamma_r(t)=re^{it}##. Prove that ##\lim_{r \to \infty} \int_{\gamma_r} \dfrac{e^{iz}}{z}dz=0##. The attempt at a solution. The only thing I could do was: ## \int_{\gamma_r} \dfrac{e^{iz}}{z}dz=\int_0^{\pi}...

\int(z^2+1)^2dz Evaluate this over the cycloidx=a(\theta-sin\theta) and y=a(1-cos\theta) for \theta =0 to \theta = 2\pi Am I on the right track, or do I need to approach this a different way? for z^2 we have (x+iy)^2, so x^2-y^2 + i2xy for the real part...

[b]1. Homework Statement Evaluate the following real integral using complex integrals: \int_0^\infty \frac{cos(2x)}{x^2+4}dx Homework Equations Cauchy's Residue Theorem for simple pole at a: Res(f;a)=\displaystyle\lim_{z\rightarrow a} (z-a)f(z)The Attempt at a Solution Since the function...

Homework Statement I have an integral of the form: ∫0∞exp(ax+ibx)/x dx What is the general method for solving an integral of this kind. Homework Equations Maybe residual calculus? The Attempt at a Solution

hello everybody I'd like to understand what mean the result of a complex integral. For example, integrate f(z) = z² from 0 to 2+i results 2/3 + 11/3 i. But, what is this? What 2/3 + 11/3 i represents geometrically? Is it possivel view this result? Thx!

I need help to solve this problem from Complex variables, Arthur A. Hauser, Ch. 5. pag. 122. Problem 5.42 show that ∫ csc(z)dz/z = 0 where C is the unit circle around the origin. Solve it without using The Cauchy Integral Formula...

Homework Statement We need to calculate this complex integral as line integral: Homework Equations The Attempt at a Solution This is correct, I guess: But not sure about this part: Are dx, dy, x, y chages correct or there is other method to use?

Hi I have been using a textbook which shows that ∫cos z/z^2 around the circle |z|=1 is zero by doing a Laurent expansion and finding the residue is zero. I was under the impression that only analytic functions have a integral of zero around a closed surface. ( the Cauchy-Goursat Theorem )...

Though it is not homework I posted this here, hopefully it'll get more action. Thanks given \int_{\Lambda} \frac{e^{i w}}{w^2 + 1} \mathrm{d}w where ##w \in \mathbb{C}## with ##Im(w) \geqslant 0## and ##|w| = \xi## want to evaluate the behavior of the Integral as ##\xi \rightarrow \infty##...

Homework Statement A semi-infinite bar (0 < x < 1) with unit thermal conductivity is fully insulated at x = 0, and is constantly heated at x = 1 over such a narrow interval that the heating may be represented by a delta function: \frac{\partial U}{\partial t}=\frac{\partial^2 U}{\partial...

Hi, i'm studing the divergent/convergent behavior of some feynman diagrams that emerge from the study of luttinger liquid. One of this diagrams has a loop inside it and loop-integrals has the following form: \int_{-\Lambda}^{+\Lambda}dQ\int...

greetings , we have the following integral : I(x)=\lim_{T\rightarrow \infty}\frac{1}{2\pi i}\int_{\gamma-iT}^{\gamma+iT}\sin(n\pi s)\frac{x^{s}}{s}ds n is an integer . and \gamma >1 if x>1 we can close the contour to the left . namely, consider the contour : C_{a}=C_{1}\cup C_{2}\cup...

Homework Statement Use Cauchy's Integral Theorem to evaluate the following integral ##\int_0^{\infty} \frac{x^2+1}{(x^2+9)^2} dx## Homework Equations Res ##f(z)_{z=z_0} = Res_{z=z_0} \frac{p(z)}{q(z)}=\frac{p(z_0)}{q'(z_0)}## The Attempt at a Solution I determine the roots of...

Homework Statement z is a complex variable. What is antiderivative of \frac{e^{-iz}}{z^2+(\mu r)^2}? Homework Equations The Attempt at a Solution To caluculate the Fourier transform encounterd in reading quantum phsycis i have to calcualte this integral. I have little...

1. let C be the circle |z| = 2 traveled once in the positive sense. Computer the following integrals... a.∫c zez/(2z-3) dz Homework Equations I am confused as to a step in my solution, but i believe a relevant equation is if i am integrating over a circle and the function is analytic...

I have been working in complex functions and this is a new animal I came across. Let γ be a piecewise smooth curve from -1 to 1, and let A=∫γa(x2-y2) + 2bxy dz B=∫γ2axy - b(x2-y2) dz Prove A + Bi = (2/3)(a-bi) In the past anything like this defined γ and I would find a parametric...

Is my solution to the following problem correct? Evaluate $$ \int_\gamma \frac{z^3}{z-3} dz$$ where $\gamma$ is the circle of radius 4 centered at the origin. Solution Form the cauchy integral formula we have that: $$ f(3)=\frac{1}{2\pi i} \int_\gamma \frac{z^3}{z-3}dz$$ and so $$...

Homework Statement Consider the integral of the function (1) around a large circle of radius R>>b which avoids the singularities of (e^{z}+1)^{-1}. Use this result to determine the sum (2) and (3). Homework Equations (1) - f(z) = \frac{1}{(z^2-b^2)(e^z+1)} (2) -...

Is there a problem with the following evaluation?\displaystyle \int e^{-ix^{2}} \ dx = \frac{1}{\sqrt{i}} \int e^{-u^{2}} \ du = \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf}(u) + C = \left(\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right) \text{erf} (\sqrt{i}x) + C So...

$\displaystyle \int_{0}^{\infty} e^{-ix^{2}} \ dx = \lim_{R \to \infty} \int_{0}^{R} e^{-ix^{2}} \ dx $ I think I'm perfectly justified in treating $i$ like a constant since we're integrating with respect to a real variable.$ \displaystyle = \frac{1}{\sqrt{i}} \ \lim_{R \to \infty}...

$\displaystyle \int_0^1 \frac{2t+i}{t^2+it+1} dt = \int_0^1 \left(\frac{t}{2} + \frac{i}{4} + \frac{5/4}{2t+i}\right) dt = \frac{1}{4} + \frac{5}{8} \ln\left(\sqrt{5}\right) + i\left(\frac{1}{4} + \frac{5}{8}\tan^{-1}\left(\frac{1}{2}\right)\right)$ Is this correct?