What is Contour integral: Definition and 122 Discussions

In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane.Contour integration is closely related to the calculus of residues, a method of complex analysis.
One use for contour integrals is the evaluation of integrals along the real line that are not readily found by using only real variable methods.Contour integration methods include:

direct integration of a complex-valued function along a curve in the complex plane (a contour);
application of the Cauchy integral formula; and
application of the residue theorem.One method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums.

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  1. Mayhem

    Can the contour integral of z⁷ be simplified using a parameterized expression?

    First I parameterize ##z## by ##z(t) = 5i + (3 + i - 5i)t## such that ##z(0) = 5i## and ##z(1) = 3 + i##, which means that ##0 \leq t \leq 0## traces the entire line on the complex plane. By distributing ##t##, we achieve a parameterized expression of the form ##z(t) = x(t) + iy(t)## $$z(t) = 3t...
  2. Euge

    POTW Contour Integral Representation of a Function

    Suppose ##f## is holomorphic in an open neighborhood of the closed unit disk ##\overline{\mathbb{D}} = \{z\in \mathbb{C}\mid |z| \le 1\}##. Derive the integral representation $$f(z) = \frac{1}{2\pi i}\oint_{|w| = 1} \frac{\operatorname{Re}(f(w))}{w}\,\frac{w + z}{w - z}\, dw +...
  3. brotherbobby

    Line integral of a scalar function about a quadrant

    Problem : We are required to show that ##I = \int_C x^2y\;ds = \frac{1}{3}##. Attempt : Before I begin, let me post an image of the problem situation, on the right. I would like to do this problem in three ways, starting with the simplest way - using (plane) polar coordinates. (1) In (plane)...
  4. D

    I Help With a Proof using Contour Integration

    I am reading a proof in Feedback Systems by Astrom, for the Bode Sensitivity Integral, pg 339. I am stuck on a specific part of the proof. He is evaluating an integral along a contour which makes up the imaginary axis. He has the following: $$ -i\int_{-iR}^{iR}...
  5. Othman0111

    Contour Integral to Properly Posting Math Problems on Online Forums

    Homework Statement Homework Equations The Attempt at a Solution
  6. Santilopez10

    How Does the Sinc Function Integral Relate to Quantum Collision Theory?

    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 &...
  7. Safder Aree

    Contour Integration over Square, Complex Anaylsis

    Homework Statement Show that $$\int_C e^zdz = 0$$ Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. Homework Equations $$z = x + iy$$ The Attempt at a Solution I know that if a function is analytic/holomorphic on a domain and the contour lies...
  8. N

    On deriving the standard form of the Klein-Gordon propagator

    I'm trying to make sense of the derivation of the Klein-Gordon propagator in Peskin and Schroeder using contour integration. It seems the main step in the argument is that ## e^{-i p^0(x^0-y^0)} ## tends to zero (in the ##r\rightarrow\infty## limit) along a semicircular contour below (resp...
  9. aphirst

    I Derivative and Parameterisation of a Contour Integral

    As part of the work I'm doing, I'm evaluating a contour integral: $$\Omega \equiv \oint_{\Omega} \mathbf{f}(\mathbf{s}) \cdot \mathrm{d}\mathbf{s}$$ along the border of a region on a surface ##\mathbf{s}(u,v)##, where ##u,v## are local curvilinear coordinates, and where the surface itself is...
  10. B

    Contour integral and problem of Quantum mechanics (Griffiths)

    Homework Statement Homework Equations This is solution of Griffith problem 11.16 The Attempt at a Solution This is procedure to get a 1-D integral form of Schrodinger equation. I don't understand why that contour integral include only one pole for each contour?
  11. M

    I Contour integral from "QFT for the gifted amateur"

    Hi, Could you please help me understand the following example from page 76 of "QFT for the gifted amatur"? I can't see how the following integral becomes Thanks a lot
  12. J

    A Is the pole in this integrand integrable?

    I am trying to numerically integrate the following complicated expression: $$\frac{-2\exp{\frac{-4m(u^2+v^2+vw+w^2+u(v+w))}{\hbar^2\beta}-\frac{\hbar\beta(16\epsilon^2-8m\epsilon(-uv+uw+vw+w^2-4(u+w)\xi...
  13. mercenarycor

    Contour integration with a branch cut

    Homework Statement ∫-11 dx/(√(1-x2)(a+bx)) a>b>0 Homework Equations f(z0)=(1/2πi)∫f(z)dz/(z-z0) The Attempt at a Solution I have absolutely no idea what I'm doing. I'm taking Mathematical Methods, and this chapter is making absolutely no sense to me. I understand enough to tell I'm supposed...
  14. F

    I Deformation of contour of integration or shifting poles

    As I understand it, in order to compute a contour integral one can deform the contour of integration, such that it doesn't pass through any poles of the integrand, and the result is identical to that found using the original contour of integration considered. However, I have seen applications...
  15. G

    A A problem about branch cut in contour integral

    Hello. I have a difficulty to understand the branch cut introduced to solve this integral. \int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]} here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
  16. dykuma

    Contour integral using residue theorem

    Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...
  17. F

    I Contour integration - reversing orientation

    I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##. Using their notation, consider a contour ##\mathcal{C}## with...
  18. Zeeree

    Finding an upper bound for a contour integral (Complex)

    C1 1. Homework Statement : Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz where the contour C = C1 + C2. C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B...
  19. P

    Contour integral- Complex variables

    Homework Statement evaluate ##\int \frac{sinh(ax)}{sinh(\pi x)}## where the integral runs from 0 to infinity Homework EquationsThe Attempt at a Solution consider ##\frac{sinh(az)}{sinh(\pi z)}## Poles are at ##z= n \pi i## So I'm considering the contour integral around the closed contour from...
  20. J

    I Need help with this definite integral

    I'm having a tough time with this integral: $$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$ where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck. What contour should I...
  21. P

    Complex Analysis: Contour Integration Question

    Homework Statement State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals. \begin{eqnarray*} (i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\ (ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...
  22. matt_crouch

    How do I solve complex contour integrals in complex analysis?

    I am trying to teach myself complex analysis . There seems to be multiple ways of achieving the same thing and I am unsure on which approach to take, I am also struggling to visualise the problem...Would someone show me step by step how to solve for example...
  23. B

    Contour Integral, Where am I going wrong?

    So, my book (Mathematical Methods in the Physical Sciences 3rd ed by Boas) proposed a problem that I have really been struggling with: I know it is probably just an algebra mistake, but I have gone over it multiple times and cannot seem to find my mistake. Any ideas? The answer is supposed...
  24. M

    Contour integral example from "QFT for the gifted amatueur"

    Hi, I've never studied compex analysis before but I am trying to understand this example from "QFT for the gifted amatur". I don't understand why the residue at the pole is e-iEp(t-t')e-e(t-t'). How did the find e-e(t-t')? Thanks.
  25. D

    Contour integration & the residue theorem

    When one uses a contour integral to evaluate an integral on the real line, for example \int_{-\infty}^{\infty}\frac{dz}{(1+x)^{3}} is it correct to say that one analytically continues the integrand onto the complex plane and integrate it over a closed contour ##C## (over a semi-circle of radius...
  26. R

    Contour Integral from Peskin & Schroeder Intro to QFT

    1. I'm having some trouble with some of the contour integrals covered in Chapter 2 of Peskin & Schroeder's Intro to QTF. I'm not so much as looking for answers to the integral (in fact, the answers are given in the textbook), but I was hoping someone could point me to some resources to use to...
  27. B

    How can the difficult Gaussian integral be solved using standard tricks?

    Hi everyone, in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like: I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} where a,b,c are...
  28. ognik

    MHB Contour integral representation of Kronecker delta

    I'm rather impressed with complex analysis, but clearly I have a lot to learn. I'm told $ \frac{1}{2\pi i} \oint {z}^{m-n-1} dz $ is a rep. of the kronecker delta function, so I tried to work through that. I used $z = re^{i\theta}$ and got to $ \frac{1}{2\pi}...
  29. D

    How Do You Solve Complex Contour Integrals with Trigonometric Substitutions?

    I am attempting to calculate the following integral. $$\frac{1}{2\pi i}\int_C \frac{du}{u^2} exp({-\frac{(q - \frac{q_0}{2i} (u - u^{-1}))^2}{2\sigma^2}})$$ Taken over the unit disk. I first make the substitution $$z = q - \frac{q_0}{2i} (u - u^{-1})$$ So, $$dz = -\frac{q_0}{2i}(1 +...
  30. N

    Complex Analysis: Contour Integral

    Here's a link to a professor's notes on a contour integration example. https://math.nyu.edu/faculty/childres/lec12.pdf I don't understand where the ##e^{i\pi /2} I## comes from in the first problem. It seems like it should be ##e^{i\pi}## instead since ##-C_3## and ##C_1## are both on the real...
  31. A

    How Can I Solve This Contour Integral with a Pole at Zero?

    I want to solve this contour integral $$J(\omega)= \frac{1}{2\pi}\frac{\gamma_i\lambda^2}{(\lambda^2+(\omega_i-\Delta-\omega)^2)} $$ $$N(\omega)=\frac{1}{e^{\frac{-\omega t}{T}}-1}$$ $$\int_0^\infty J(\omega)N(\omega)$$ there are three poles I don't know how I get rid of pole on zero (pole in...
  32. M

    Evaluating this contour integral

    I was reading a paper which featured the following horrendous integral ##\displaystyle\prod_{n=1}^L\oint_{C_n}\frac{dx_n}{2\pi i}\prod_{k<l}^L(x_k-x_l)\prod_{m=1}^L\frac{Q_w(x_m)}{Q^+_\theta(x_m)Q^-_\theta(x_m)}## where ##Q^\pm_\theta(x)=\prod_{k=1}^L(u-\theta_k\pm \frac{i}{2})## and...
  33. P

    Integrating a Photon Gas: Contour Integration for the Grand Potential

    When considering the grand potential for a photon gas, one encounters an integral of the form: \Sigma = a\int_{0}^{\infty}x^2\ln\Big(1 - e^{-bx}\Big)dx I have never had to integrate something like this before; I was told it is done via contour integration, but I have never used such a method...
  34. A

    Complex Contour Integral Problem, meaning

    Homework Statement First, let's take a look at the complex line integral. What is the geometry of the complex line integral? If we look at the real line integral GIF: [2]: http://en.wikipedia.org/wiki/File:Line_integral_of_scalar_field.gif The real line integral is a path, but then you...
  35. M

    Contour integral with a pole on contour - justification?

    Hi everyone. While solving different contour integrals, I stumbled upon quite a few line integrals with pole(s) on contour. I've always solved them the same way, using the rule that for such lines the integrals equal \int\limits_\gamma f(z)dz=\pi i\sum\limits_n Res(f,p_n), where p_n are the...
  36. E

    Evaluating a Contour Integral of log(z) | Explaining Branch Cuts

    Is it possible to evaluate the integral of log(z) taken over any simple closed contour encircling the origin? I don't fully understand how singularities on branch cuts should be treated when integrating over contours encircling such singularities. Are residues applied? Can someone explain this...
  37. C

    Contour integral involving gamma function

    Homework Statement Evaluate the integral by closing a contour in the complex plane $$\int_{-\infty}^{\infty} dx e^{iax^2/2}$$ Homework Equations Residue theoremThe Attempt at a Solution My initial choice of contour was a semicircle of radius R and a line segment from -R to R. In the limit R to...
  38. H

    A contour integral frequenctly encountered

    Homework Statement Need some help here on a frequently encountered integral in Green's function formalism. Homework Equations I have an integral/summation as a product of a retarded and advanced Green's functions, looks simply like...
  39. M

    Contour Integral of |z| = 2 using Cauchy's Formula

    Homework Statement |z| = 2, \oint\frac{1}{z^3} Homework Equations Cauchy's Integral Formula http://en.wikipedia.org/wiki/Cauchy%27s_integral_formula The Attempt at a Solution Seems like a simple application of the general formula on the wiki page with n = 2, a = 0, and f(z) = 1...
  40. D

    How Do You Solve a Complex Contour Integral with a Non-Standard Path?

    Hi I'm really not sure how to start this question. I could do it if it was in terms of z but I'm not sure if trying to change the variable using z = x + iy is correct. If anyone could suggest a method I'd appreciate it. ∫(x3 - iy2)dz along the path z= \gamma(t) = t + it3, 0≤t≤1 Thanks
  41. F

    MHB How do I evaluate the contour integral for $f'(z)/f(z)$ around $|z|=4$?

    Let $f(z)=\frac{(z^2+1)^2}{(z^2+2z+2)^3}$ . Evaluate the contour integral of $f'(z)/f(z)$ around the circle $|z|=4$? How do I do this without having to find $f'(z)$? Thanks
  42. F

    Complex contour integral with a second order pole at origin

    Homework Statement Hello all. I'm currently attempting to prove the central limit theorem using a simple case of two uniformly distributed random variables. Aside from being able to solve it using convolutions, I also wish to solve it by using the Dirac Delta function. That aside, the integral...
  43. M

    Contour integral of a complex number

    Hello. I am stuck at the third point, that is from 1+i to i. I asked someone to show me his answer but that part of his is different from mine. Is his solution correct? Here it is: (i) z = 0 to 1 via z(t) = t with t in [0, 1]: ∫c1 Re(z^2) dz = ∫(t = 0 to 1) Re(t^2) * 1 dt = ∫(t = 0 to 1)...
  44. A

    MHB Contour integral (please check my solution)

    Hello. Can someone check my solution for this question? I am not sure what to do about the "from e^-pi*1/2 to e^pi*i/2" part. I ignored that part.
  45. N

    Complex contour integral zero while containing a pole?

    [SOLVED] Complex contour integral zero while containing a pole? Homework Statement ##f(z) = \frac{1}{z^2 +2z +5} = \frac{1}{(z-z_1)(z-z_2)}##, where ##z_1= -1+2i## and ##z_2 = -1-2i##. Now, let z be parametrized as ##z(\theta)=Re^{i \theta}##, where ##\theta## can have values in the...
  46. ShayanJ

    Contour integral with poles on contour

    In the process of calculating the integral \int_0^{2\pi}\frac{\sin{x} \cos{x}}{\sin{x}+\cos{x}}dx by contour integration,I got the following: -\frac{1}{2}[ \LARGE{\oint} \large{\frac{z^2}{(1-i)z^2+i+1}}dz-\LARGE{\oint}\large{\frac{z^{-2}}{(1-i)z^2+i+1}}dz] Where the contour of integration...
  47. M

    Contour Integral Homework: Get the Answer

    Homework Statement Which one is correct? https://mail.google.com/mail/u/0/?ui=2&ik=f891924403&view=att&th=1429c0d877bc3684&attid=0.1&disp=safe&realattid=1452904307646005248-1&zw Homework Equations The Attempt at a Solution
  48. D

    Contour integral along the imaginary axis

    I'd like to evaluate the integral, \int^{i\infty}_{-i\infty} \frac{e^{iz}}{z^2 + a^2}dz along the imaginary axis. This function has poles at z = \pm ia , with corresponding residues \textrm{res}(\frac{e^{iz}}{z^2 + a^2},\pm ia) = \pm\frac{e^{\mp a}}{2ai} My question is - I'm not sure...
  49. J

    How to derive beta function as pochhammer contour integral?

    Hi, We have: \beta(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}dt,\quad Re(a)>0, Re(b)>0 and according to Wikipedia: http://en.wikipedia.org/wiki/Pochhammer_contour we can write: \left(1-e^{2\pi ia}\right)\left(1-e^{2\pi ib}\right)\beta(a,b)= \int_P t^{a-1}(1-t)^{b-1}dt valid for all a and b where P...
  50. ShayanJ

    How can I calculate this integral using contour integration?

    I want to calculate the integral \int_0^{\infty} \frac{x^a}{(1+x)^2}dx \ (-1<a<1) via contour integration But it seems a little tricky. I tried to solve it like example4 in the page ( http://en.wikipedia.org/wiki/Contour_integral#Example_.28IV.29_.E2.80.93_branch_cuts ) but I arrived at zero...