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

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)...

7. ### 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. ### 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. ### 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. ### 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. ### 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

30. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...