Contour integral Definition and 116 Threads

Homework Statement Using contour integration, evaluate ##\int_{0}^{\infty} \frac{\sqrt{x}}{x^3 + 1} dx##. The Attempt at a Solution Normally what I try to do in these problems is consider the upper half of a semi-circle from -R to R in the complex plane, as R goes to infinity. In doing...

Hi there! I'm almost sure that somebody has previously make this same question so, if it is like that, I'm sorry. I've just been introduced to contour integrals, I've tried to look around the internet and some textbooks, but i can't find out what do they actually are so, if someone could...

Homework Statement What is the integral of e-1/z around a unit circle centered at z = 0? Homework Equations - The Attempt at a Solution The Laurent expansion of this function gives : 1 - 1/z + 1/(2 z^2) - 1/(3! z^3) + . . . . . The residue of the pole inside is -1. So the integral...

Hi I am struggling trying to see understand the basic propagator integral trick. \int \frac{d^{3}p}{(2\pi^{3})}\left\lbrace \frac{1}{2E_{p}}e^{-ip.(x-y)}|_{p_{0}=E_{p}}+\frac{1}{-2E_{p}}e^{ip.(x-y)}|_{p_{0}=-E_{p}}\right\rbrace = \int \frac{d^{3}p}{(2\pi^{3})}\int...

Consider the function $$f(z)=Log(\frac{z-a}{z-b})$$ where $a,b\in D(0,r)$ , the disc of radius $r$ centered at the origin, open, and $r>0$. Show that $f$ is holomophic in an open subset of complex plane, containing the complement of the disc:$\mathbb C-D(0,r)$ and compute the integral $$...

I tried posting this at stack exchange but it never got the question answered. I want to prove this: If f:U→C is holomorphic in U and invertible, P∈U and if D(P,r) is a sufficently small disc about P, then f^{-1}(w) = \frac{1}{2\pi i} \oint_{\partial D(P,r)}{\frac{sf'(s)}{f(s)-w}}ds

Homework Statement Evaluate the integral: \int^{2\pi}_{0}\frac{d\theta}{(A+Bcos(\theta))^2} a^2>b^2 a>0 The Attempt at a Solution First, I convert this to contour integration along a full sphere in the complex plane. I let: z=e^(i\theta) dz=ie^(i\theta) d\theta=-idz/z...

Homework Statement I have the following problem: Compute \operatorname{Re} \int _\gamma \frac{\sqrt{z}}{z+1} dz, where \gamma is the quarter-circle \{ z: |z|=1, \operatorname{Re}z \geq 0 , \operatorname{Im} z \geq 0 \} oriented from 1 to i, and \sqrt{z} denotes the principal...

You can do integrals of real functions like: \oint1/(3-sinθ) by transforming to a complex contour, which enloses the origin, and then using the residue theorem. Normally you would transform to the unit circle, but in principal you could use any contour (right?). Now, sometimes you find that...

Homework Statement Evaluate \oint\frac{1}{a-cos\theta} d\theta from [0:2\pi] Homework Equations Method of contour integration. The Attempt at a Solution I make the substitution z=exp(i\theta) \Rightarrow giving me the integral: \oint1/(a-½(z+z-1)) dz/iz from [0:2\pi] And...

Trying to teach myself contour integration, but I'm not so good at it. I want help with evaluating the closed integral: ∫sinθ/(a-sinθ)dθ from -\pi to \pi So I substitute z= eiθ, and sinθ = -i/2(z-z-1) and dθ = -ie-iθdz So our integral becomes: ∫-i/2(z-z-1)/(a+i/2(z-z-1) dz Is this...

Homework Statement Convert the following to an equivalent cotour integral around |z|=1 then use Cauchy's integral formula to evaluate it. ##\int_{0}^{2 \pi} \frac {d \theta}{13+5 \sin \theta}## Homework Equations let ##z=e^{i \theta}## The Attempt at a Solution ##d \theta =...

Homework Statement Evaluate each of the following by Cauchy's Integral formula a)## \int_cj \frac{\cos z}{3z-3\pi} dz## c1: |z|=3, c2:|z|=4 b) ##\int_c \frac{e^{3z}}{z-ln(2)} dz## c=square with corners at ##\pm(1\pm i)## Homework Equations ##f(z_0)=\frac{1}{2 \pi i}\int_c...

Homework Statement Folks, How do I evaluate the integral of (z+2)/z dz for the path C= the top half of the circle |z|=2 from z=2 to z=-2. The Attempt at a Solution I take ##z=x+iy## and ##dz=dx+idy## Therefore ##\int_c f(z)=\int_c (1+(2/(x+iy))(dx+idy)##...not sure if I'm going the...

Homework Statement 1. Evaluate \int_{c_{2}(0)} f(z)dz = \int_{c_{2}(0)} \frac{z^{m}}{1+z^{3}}dz Where c_{2}(0) is the circle of radius 2 centered at the origin with positive orientation (ccw). I have done the question myself and compared it with the solution. However, I don't think I am...

Homework Statement Find the contour integral of Log(z). The contour is defined as: x^2 + 4y^2 = 4, x>= 0, y>=0 Homework Equations The Attempt at a Solution parametrize the contour as z(t) = 2cos(t) + isin(t) 0 <= t <= pi/2 The contour integral = ∫Log(z(t))z'(t)dt I am having...

If C is a simple closed contour such that w lies interior to C, and n > 1, then \int_{C} \frac{dz}{(z-w)^n} = 0. I'm confused because the function f(z) = (z-w)^{-n} has a pole at w, so it isn't holomorphic, but the integral is still zero. The Cauchy-Goursat Theorem says that if f is...

Suppose I want to compute tthe integral: \int_{-\infty}^{\infty}\frac{\textrm{sech}\hspace{0.1cm} x}{x^{2}-2|x|+1}dx Can I compute this integral via contour integration? The only way that I have thought of is to split up the domain: \int_{-\infty}^{\infty}\frac{\textrm{sech}\hspace{0.1cm}...

Homework Statement The Attempt at a Solution We can parametrise the contour \gamma (the positively oriented unit circle) by \gamma(t) = e^{it} for t \in [0, 2\pi ] So by the definition of a contour integral \displaystyle I = \frac{1}{2\pi i} \int^{2\pi}_0...

I've tried this using the definition of a contour integral and Cauchy's residue theorem but get conflicting answers. \displaystyle f(z) = \frac{z+1}{z(2z-1)(z+2)} We can parametrise the contour \gamma (the unit circle) by \gamma(t) = e^{it} for t \in [0, 2\pi ] So by the definition of a...

Homework Statement Use the contour integral \int_{C}\frac{e^{pz}}{1+e^z}dz to evaluate the real integral \int^{\infty}_{- \infty}\frac{e^{px}}{1+e^x}dx 0<p<1 The contour is attached. It is a closed rectangle in the positive half of the complex plane. It height is 2i∏. Homework...

For my research I am trying to solve a particular integral. I know that I have to transform it properly to use the residue theorem, but I am having difficulty attaining this form. Here is the integral: \int^{\infty}_{-\infty}\frac{e^{ix}dx}{\sqrt{a - x^2 + ib}} where a and b are real...

Homework Statement The integral I'm trying to solve is sqrt(x)/(1+x^2) from 0 to infinity. Homework Equations The Attempt at a Solution I've attached my solution. I know it's not right, as I shouldn't get an imaginary solution. The answer is actually pi/sqrt(2) according to my...

Homework Statement I am reading the solution the integral of (log(z))^2/(1+z^2) from 0 to infinity in a textbook, and I'm not sure I quite understand it, and I think this misunderstanding stems from my difficulty w/ branch points/cuts for multivalued functions. Homework Equations...

Homework Statement I am to integrate cos(2x)/(x-i*pi) from -inf to inf Homework Equations The Attempt at a Solution The problem I'm having is this: I write cos(2x) in exponential form, e^(2iz), so f(z) = e^(2iz)/(z-i*pi). I choose a large semicircle in the UHP as my contour...

I want to perform the following integral which comes from inverting a Laplace transform: \lim_{R\rightarrow\infty}\int_{\sigma -Ri}^{\sigma +Ri}\frac{e^{sx}}{\sqrt{s^{2}-a^{2}}}ds Would it be some kind of keyhole contour? Mat

Hello, I am working on a problem and I have to solve a nasty integral. The problem is that I am not sure if the method I am using is correct. The integral I need to solve is: \int^{\infty}_{-\infty}\frac{e^{ikx}dk}{\sqrt{k^2 + a} - (b\pm i\lambda)} At this point, I have tried multiplying the...

Homework Statement I have to prove the following: \int_0^{2\pi} \frac{\mathrm{d}\theta}{(a + cos(\theta))^2} = \frac{2pia}{(a^{2}-1)^{3/2}} for a > 1. Homework Equations I have an example at hand for \int_0^{2\pi} \frac{\mathrm{d}\theta}{a + cos(\theta)} from which I know I have to...

Homework Statement Let C be a contour formed by the points O(0,0), A(1,0), B(1,1), with the direction OA->AB->BO. By using the definition of a contour integral, evaluate: (integral) f(z)dz Homework Equations \int f[z(t)]z'(t)dt The Attempt at a Solution I didn't include the work I've done...

I want to compute kind of following problems. int(from 0 to infinity) e^(-x) / (x-1) dx= I using contour integrals, then 2 pi i I = -pi i / (2) Res[e^(-x) ln(x) /(x-1) , x = 1] - pi i / (2) Res[e^(-x) /(x-1) (ln(x) + 2 pi i) , x = 1] I = e^(-1) / (2) pi i I know there is some...

Homework Statement Consider I = \int_0^{\infty} dx \frac{\mathrm{ln}(x)}{x^a(1+x)}, 0<a<1. a) Calculate \oint dz \frac{\mathrm{ln}(z)}{z^a(1+z))}, along a keyhole contour. b) Split the contour integral into several parts and calculate these parts separately. Compare to the result of (a) and...

Homework Statement I have to evaluate the following integral by means of the Cauchy Integral Theorem: \int_{- \infty}^{\infty}\frac{e^{-ikx}}{(x+i)(x+2i)}dxHomework Equations f(z_0)=\frac{1}{2 \pi i} \oint_C \frac{f(z)}{z-z_0}dzThe Attempt at a Solution The idea I had was to consider the...

Contour integral with multiple singularities inside domain without residue theorem?? Homework Statement Evaluate \oint\frac{dz}{z^{2}-1} where C is the circle \left|z\right| = 2 Homework Equations Just learned contour integrals, so not much. Ok to use Cauchy's Integral formula (if...

Homework Statement The following function : a) f(z) = \frac{1}{z^6 + 1} has simple poles on : z_1 = e^{pi/6 i}, z_2 = e^{3pi/6 i}, z_3 = e^{5pi/6 i} I know how to get the poles, but how could I demonstrate they are simple (order 1) ? I tried to write the Laurent series...

Hi, During my research I came across a contour integral where the pole was on the boundary. I have never come across this before, do anyone of you know how I would go about computing this? It involved the Hilbert transform and I can't find it in my undergraduate complex analysis books and...

Homework Statement The Attempt at a Solution [SIZE="4"]PART1) Series of Taylor for e^(3z) + 2 (based on series for e^z) \sum_{n=0}^{\infty} \frac{(3z)^{n}}{n!}= (1+2) + 3z + \frac{(3z)^2}{2!} + \frac{(3z)^3}{3!} + ... Series of Taylor for 3e^(z) (based on series for e^z)...

So the length of the contour is L(gamma) = 2.pi and so i have http://images.planetmath.org:8080/cache/objects/7138/js/img1.png so i need to show max f(z) = e? So the maximum of f(z)= e1/z2 in the unit circle centre 0, radius 1 implies that 1/z2 should be maximum, and this is when z2 is its...

Homework Statement Integrate exp(-z^2) over the rectangle with vertices at 0, R, R + ia, and ia. Homework Equations int(0, inf)(exp(-x^2)) = sqrt(pi/2) The Attempt at a Solution I really don't have much of an idea here - the function is analytic so has no residues... The...

Homework Statement let g denote the elliptic arc parametrized by z(t) = 2cost + 3isint, for t between 0 and pi/2 (inclusive). Evaluate the integral of f(z) = z[sin(pi*z^2) - cos(pi*z^2)] over g. Homework Equations If g is determined by the function z mapping from [a,b] to C and...

Homework Statement Calculate the following integral along three different circular contours, \int_{C_j}\frac{dz}{z(3z-1)^2(z+2)} where C_1:0<r_1<1/3 C_2:1/3<r_2<2 C_3: r_3>2 The Attempt at a Solution The function has singularities at z=0, z=1/3 and z=-2. Thus all three contours enclose...

Homework Statement Calculate the following line integrals from point z'=(0,-1) to z"=(0,1) along three different contours, C_j=(0,1,2). \int_{C_j}|z|dz where C_0 is the straight line along the y-axis, C_1 is the right semi-circular contour of radius 1, and C_2 is the left semi-circular...

I saw a contour integral in a text I was recently reading, but unfortunately a contour integral is beyond my understanding at the moment. As such, I would greatly appreciate it if someone could explain a contour integral to me. If it helps, I know about derivatives, integrals, partial...

Homework Statement V(x) = \int \frac{d^3q}{(2\pi)^3} \frac{-g^2}{|\vec{q}|^2 + m^2} \exp^{i \vec{q} \cdot \vec{x}} = -\frac{g^2}{4\pi^2} \int_0^{\infty} dq q^2 \frac{exp^{iqr}-exp^{-iqr}}{iqr} \frac{1}{q^2+m^2} = \frac{-g^2}{4\pi^2 i r} \int_{-\infty}^{\infty} dq \frac{q...

Homework Statement Compute \int_{\alpha}^{\beta}{\left(\frac{\beta - x}{x-\alpha}\right)^{a-1} \frac{dx}{x}} where 0 \leq a \leq 2 and 0 \leq \alpha \leq \beta . Homework Equations Cauchy's theorem, Residue theorem The Attempt at a Solution I'm confused about setting this up...

Homework Statement Show that: For a = 0 \int_{0}^{\infty} \frac{cos{ax}+x sin{ax}}{1+x^2} dx = \frac{\pi}{2} For a > 0 \int_{0}^{\infty} \frac{cos{ax}+x sin{ax}}{1+x^2} dx = \pi e^{-a} For a < 0 \int_{0}^{\infty} \frac{cos{ax}+x sin{ax}}{1+x^2} dx = 0 Homework Equations Residue...

Homework Statement describe the laurent series for the function f(z) = z^3 cos(\frac {1}{z^2}) b) use your answer to part a to compute the contour integral \int z^3 cos(\frac {1}{z^2}) dz where C is the unit counter-clockwise circle around the origin.Homework Equations The Attempt at a...

Homework Statement let t be the triangle with vertices at the points -3, 2i, and 3, oriented counterclockwise. compute \int \frac {z+1}{z^2 + 1} dz Homework Equations f(z) = \frac {1}{2 \pi i} * \int \frac {f(z)}{z - z_o} dz The Attempt at a Solution the integrand fails to be analytic at...

Hi there! I am trying to prove the following 2 identities using complex analysis methods and contour integration and I'm really stuck on defining the integration paths. \int_{0}^{1}\frac{\log(x+1)}{x^2+1}d x=\frac{\pi\log2}{8} \int_{0}^{\infty}\frac{x^3}{e^x-1}d x=\frac{\pi^2}{15}...

Homework Statement http://img243.imageshack.us/img243/4339/69855059.jpg I can't seem to get far. It makes use of the Exponentional Taylor Series: Homework Equations http://img31.imageshack.us/img31/6163/37267605.jpg The Attempt at a Solution taylor series expansions for cos...

Is there a way to perform a contour integral around zero of something like f(z)/z e^(1/z), where f is holomorphic at 0? If you expand you get something like: \frac{1}{z} \left( f(0) + z f'(0) + \frac{1}{2!} z^2 f''(0) + ... \right) \left( 1 + \frac{1}{z} + \frac{1}{2!} \frac{1}{z^2} + ...