What is Complex integration: Definition and 77 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. U

    Complex Integration Along Given Path

    From plotting the given path I know that the path is a curve that extends from z = 1 to z=5 on the complex plane. My plan was to parametrize the distance from z = 1 to 5 as z = x, and create a closed contour that encloses z=0, where I could use Cauchy's Integral Formula, with f(z) being 1 / (z +...
  2. LCSphysicist

    Complex integration is giving the wrong answer by a factor of two

    $$\int_{0}^{2\pi } (1+2cost)^{n}cos(nt) dt$$ $$e^{it} = z, izdt = dz$$ $$\oint (1+e^{it}+e^{-it})^{n}\frac{e^{nit}+e^{-nit}}{2} \frac{dz}{iz} = \oint (1+z+z^{-1})^{n}\frac{z^{n}+z^{-n}}{2} \frac{dz}{iz}$$ $$\oint (z+z^{2}+1)^{n}\frac{z^{2n}+1}{z^{2n+1}} \frac{dz}{2i} = \pi Res = \pi...
  3. minimoocha

    MHB Solve Complex Integration: Find 2.36 Area of y=-x/2e+1/e+e & y=e^x/2

    The area of two lines that I need to find is 2.36, however i need this in exact form. The lines are y=-x/2e+1/e+e the other line is y=e^x/2 Since y=-x/2e+1/e+e is on top it is the first function. A=(the lower boundary is 0 and the top is 2) -x/2e+1/e+e-e^x/2 If you could please help!
  4. T

    Help with a complex integration in QFT

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  5. T

    Evaluating a complex integral

    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...
  6. J

    Problem with Complex contour integration

    Homework Statement I want to compute ##I=\int_C \dfrac{e^{i \pi z^2}}{sin(\pi z)}##, where C is the path in the attached figure (See below). I want to compute this by converting the integral to one whose integration variable is real.Homework Equations There are not more relevant equations. The...
  7. Math Amateur

    MHB Complex Integration - Conway - First Example on page 63 .... Section 1, Ch. IV .... ....

    I am reading John B. Conway's book, "Functions of a Complex Variable I" (Second Edition) ... I am currently focussed on Chapter IV: Complex Integration ... Section 1: Riemann-Stieljes Integral ... ... I need help in fully understanding the first example on page 63 ... ... The the first...
  8. S

    I Learning Complex Integration: Endpoints & Paths

    Hello! I started learning about complex analysis and I am a bit confused about integration. I understand that if we take different paths for the same function, the value on the integral is different, depending on the path. But if we use the antiderivative...
  9. F

    Complex integration on a given path

    Homework Statement Calculate the following integrals on the given paths. Why does the choice of path change/not change each of the results? (a) f(z) = exp(z) on i. the upper half of the unit circle. ii. the line segment from − 1 to 1. Homework Equations ∫γf(z) = ∫f(γ(t))γ'(t)dt, with the...
  10. Macykc2

    Can I Use Antiderivatives to Evaluate this Complex Integral?

    Homework Statement I need to evaluate the following integral using the antiderivative: $$\int log^2(z) \, dz$$ I don't know how to make a subscript for the integral sign, there should be a "c" on the bottom part. C is any contour from ##π## to ##i##, not crossing the non-positive x-axis...
  11. 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...
  12. arpon

    Complex Integration using residue theorem

    Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
  13. M

    Complex integration, possibly branch cut integral

    Homework Statement The integral I want to solve is $$ D(x) = \frac{-i}{8\pi^2}\int dr\,d\theta \frac{e^{-irx\cos\theta}}{\sqrt{r^2+m^2}}r^2\sin\theta$$ which I've reduced to $$ D(x) = \frac{-i}{4\pi x}\int dr \frac{r\sin(rx)}{\sqrt{r^2+m^2}} $$ by integrating over ##\theta##. However, I...
  14. Brandon Trabucco

    B Complex Integration By Partial Fractions

    Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...
  15. Calpalned

    Calculating Arc Length for Parametric Equation x = e^t + e^-t and y = 5 - 2t

    Homework Statement The question involves finding the arc length of the parametric equation x = e^t + e^-t and y = 5 - 2t Homework Equations Arc length of a parametric equation ∫√(dy/dt)^2 + (dx/dt)^2 dt limits are from 0<t<3 The Attempt at a Solution Taking the derivative of both x and y...
  16. B

    How to think about complex integration

    Let's say you integrate a complex function along a curve. How do you visualize it? This is explaned very well in multivariate calculus in terms of work, or for instance the weight of the line of we integrate over the density etc.. But when we look at complex function I get this: The function...
  17. A

    MHB Complex Integration: Solving $\int_{|z|=1} |z-1|.|dz|$

    Can you check my work please, Compute $\displaystyle \int_{|z|=1} |z-1| . |dz| $ $ z(t) = e^{it} , 0 \leq t < 2 \pi $ $ |dz| =| ie^{it} dt | = dt $ $\displaystyle \int_{0}^{2\pi} |\cos(t) + i\sin(t) - 1 | dt $ $\displaystyle \int_{0}^{2 \pi} \sqrt{(\cos(t) -1)^2 + \sin ^2( t)} \, dt =...
  18. stripes

    Complex Integration Homework - Part (a) and (b) Help

    Homework Statement Homework Equations The Attempt at a Solution I did part (a) which is pretty easy. Use the Cauchy integral formula. Part (b) says "hence", which leads me to believe that part (a) can be used for part (b), but I cannot see anything remotely related between the...
  19. stripes

    Complex Integration Homework: Answers and Guidance

    Homework Statement Homework Equations The Attempt at a Solution I did (i) by breaking the integrand into partial fractions and then using the Cauchy Integral Formula for each integral. I got the correct answer. What does (ii) even mean? WHat does it mean to integrate "around...
  20. A

    MHB Is $f(z) = \sin(z)/z$ an analytic function on the complex plane?

    Find the integral $\displaystyle \int_C \dfrac{\sin(z)}{z} dz $ where $c: |z| = 1 $ Can I use Cauchy integral formula since sin(z) is analytic $\displaystyle\int_C \dfrac{\sin(z)}{z} dz = Res(f,0) = 2\pi i \sin(0) = 0$ I tired to compute it without using the formula $z(t) = e^{it} ...
  21. P

    Complex integration and residue theorem.

    Hi, Homework Statement I was wondering whether any of you could kindly explain to me how the equation in the attachment was derived. I mean, how could I have known that it could be separated into these two fractions? Homework Equations The attachment also specifies the integration to be...
  22. D

    MHB Solving Complex Integration Problems with \((*) Formula

    Consider \[ \int_{-\infty}^{\infty}\frac{e^{iax}}{x^2 - b^2}dx \] where \(a,b>0\). The poles are \(x=\pm b\) which are on the x axis. Usually, if the poles are on the x axis, I use that the integral is \[ 2\pi i\sum_{\text{UHP}}\text{Res} + \pi i\sum_{\text{x axis}}\text{Res}\quad (*) \]...
  23. C

    Complex integration (Using Cauchy Integral formula)

    Homework Statement $$\int_\gamma \frac{\cosh z}{2 \ln 2-z} dz$$ with ##\gamma## defined as: 1. ##|z|=1## 2. ##|z|=2## I need to solve this using Cauchy integral formula. Homework Equations Cauchy Integral Formula The Attempt at a Solution With ##|z|=2## I've solved already, as it is...
  24. V

    Complex Integration Function with multiple poles at origin

    Hello, I hope somebody can help me with this one. Homework Statement I want to find the integral of 1/x^N*exp(ix) from -inf to inf. Homework Equations It is very likely that this can somehow be solved by using Cauchy's integral formula. The Attempt at a Solution I tried to...
  25. A

    Complex integration over a curve

    Homework Statement Compute ∫C (z+i)/(z3+2z2) dz Homework Equations C is the positively orientated circle |z+2-i|=2 The Attempt at a Solution I managed to solve a similar problem where the circle was simply |z|=1, with the centre at the origin converting it to z=eiθ with 0≤θ2∏. I'm...
  26. S

    Complex Integration over a Closed Curve

    (a) Suppose \kappa is a clockwise circle of radius R centered at a complex number \mathcal{z}0. Evaluate: K_m := \oint_{\kappa}{dz(z-z_0)^m} for any integer m = 0, \pm{1},\pm{2}, ,... Show that K_m = -2\pi i if m = -2; else : K_m = 0 if m = 0, \pm{1}, \pm{2}, \pm{3},... Note...
  27. T

    Trigonometric Laurent Series and Complex Integration

    My task is to solve the integral \frac{1}{\cos 2z} on the contour z=|1| using a Laurent series. The easy part of this is the geometric part. I drew a picture of the problem. I see that there are two singularity points that occur within the contour region at \pm \frac{\pi}{4}. I realize...
  28. K

    When complex integration depends on the method of evaluation

    Homework Statement It is about an example from Essential Mathematical methods for physicists by Weber & Arfken, which describes a scattering process: I(\sigma)=\int^{+\infty}_{-\infty}\frac{x \sin x dx}{x^2-\sigma^2}2. The attempt at a solution The straightforward way is to contruct a contour...
  29. N

    Complex Integration with a removable singularity

    Hi, I'm trying to make headway on the following ghastly integral: \int_0^{\infty} x^{\frac{3}{2}}e^{-xd} J_o(rx) \frac{\sin (\gamma \sqrt{x}\sqrt{x^2+\alpha^2}t)}{\sqrt{x^2+\alpha^2}}\ dx where d,r, \alpha, \gamma ,t \in \mathbb{R}^+ and J_o is the zeroth order Bessel function of...
  30. C

    Complex Integration: Is Path Dependent?

    Homework Statement Is the integral ∫z* dz from the point (0,0) to (3,2) on the complex plane path dependent? Homework Equations I = ∫ f(z)dz = ∫udx - vdy + i ∫ vdx + udy z = x-iy, u = x, v = -y The Attempt at a Solution I have no idea how to start. The methods given in the...
  31. H

    Complex Integration: Integrating x^0.5/(1+x^2)

    Homework Statement integrate x^0.5/(1+x^2) by using complex integration Homework Equations residue theorem The Attempt at a Solution my attempt at a solution is attached.i need help in finding where am i mistaken. thank's Hedi
  32. H

    Solving Complex Integration Homework

    Homework Statement Homework EquationsThe Attempt at a Solution
  33. H

    Solving Complex Integration with Residue Theorem

    Homework Statement use residue theorem to integrate sinh(ax)/sinh(xpi) from -infinity to +infinity, a is between -pi and pi Homework Equations residue theorem 3 . The attempt at a solution i tried rectangular trajection through 0 and ia/pi with the function sinh(az)/sinh(zpi) and...
  34. Sudharaka

    MHB Math Help Forum: Solving Complex Integration Problem

    Samantha128's question from Math Help Forum, Hi Samantha128, I hope you want to show, \(\displaystyle\lim_{R\rightarrow \infty}\oint_{c}f(z)\,dz=0\). For this let us first find, \(\displaystyle\oint_{c}f(z)\,dz\) \[f(z) = \frac{z^2 + 2z -5}{(z^2+4)(z^2+2z+2)}\] The points where the...
  35. K

    Solving Complex Integration Involving Bessel, Singularities

    Well, here it is. I am at a loss as to how to approach this. I understand I can use the residue theorem for the poles at a and b, those are not the problem. I have heard that you can expand the function in a Laurent series and look at certain terms for the c term , but I don't fully understand...
  36. K

    Complex Integration - Poles on the Imaginary axis

    Homework Statement evaluate the integral: I_1 =\int_0^\infty \frac{dx}{x^2 + 1} by integrating around a semicircle in the upper half of the complex plane. Homework Equations The Attempt at a Solution first i exchange the real vaiable x with a complex variable z & factorize...
  37. D

    MHB Complex integration no Residue Theory everything else is ok

    $$ \int_0^{2\pi}\frac{\bar{z}}{z^2}dz $$ How would this be integrated?
  38. F

    Integrate Complex Function w/o Cauchy's or Residuals

    Without using Cauchy's Integral Formula or Residuals, I am trying to integrate \int_{\gamma}\frac{dz}{z^2+1} Around a circle of radius 2 centered at the origin oriented counterclockwise. \frac{i}{2}\left[\int_0^{2\pi}\frac{1}{z+i}dz-\int_0^{2\pi}\frac{1}{z-i}dz\right] \gamma(t)=2e^{it}...
  39. D

    MHB Circle radius 2 complex integration

    Gamma is a circle of radius 2 oriented counterclockwise. $$ \int_{\gamma}\frac{dz}{z^2+1} = \int_{\gamma} = \frac{i}{2}\left[\int_{\gamma}\frac{1}{z+i}dz-\int_{\gamma}\frac{1}{z-i}dz\right] $$ $\gamma(t) = 2e^{it}, \ \ \gamma'(t) = 2ie^{it}$ $$ \int_{\gamma}\frac{2ie^{it}}{2e^{it}+i}dz $$...
  40. A

    MHB Evaluating Complex Integration I_c |z^2|

    How can I evaluate I_c |z^2|,where I is the integral and c is the square with vertices at (0, 0), (1, 0), (1, 1), (0, 1) traversed anti-clockwise...?
  41. D

    MHB Complex Integration: Solving $\int_0^1\frac{2t+i}{t^2+it^2+1}dt$

    $\displaystyle\int_0^1\frac{2t+i}{t^2+it^2+1}dt = \int_0^1\frac{2t^3+3t+i-it^2}{t^4+3t^2+1}dt =\int_0^1\frac{2t^3+3t}{t^4+3t^2+1}dt+i\int_0^1 \frac{1-t^2}{t^4+3t^2+1}dt$ I tried multiplying through by the conjugate but that didn't seem fruitful and left me with the above expression. Is there a...
  42. H

    Integrate $\Sigma \frac{1}{n!}\int z^{n}e^{1/z}dz$

    Homework Statement Integrate $\Sigma \frac{1}{n!}\int z^{n}e^{1/z}dz$ Homework Equations The Attempt at a Solution Wrote out the first couple of terms, with $\frac{1}{z}=w$, making the integral $\Sigma \frac{1}{n!} (-w^{2-n}e^{w}+(2-n)(w^{1-n}e^{w}+(1-n)(w^{1-n}e^w)-(1-n)^2(w^(-n)e^w)...)$...
  43. K

    Complex integration problem using residues .

    Homework Statement I =\int \frac{cosx}{x^{2}-2x+2}dx the integral runs from -inf to inf evaluate the integral using the calculus of residues. Homework Equations shown in my attempt The Attempt at a Solution Re \oint\frac{e^{iz}}{z^{2}-2z+2} with singularities at...
  44. B

    Complex integration via parametrization

    Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction using parametrization. Show all work...
  45. M

    Complex Integrals: Sketching Paths & Computing Integrals

    Homework Statement Sketch the C1 paths a: [0; 1] -> C, t -> t + it2 and b: [0; 1 + i]. Then compute the following integrals. ∫Re(z)dz over a ∫Re(z)dz over b Homework Equations The Attempt at a Solution Sketching a seems ok, y-axis is Imaginary, x-axis is Real, and the...
  46. B

    Complex integration over a square contour (part b)

    Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction using parametrization. Show all work...
  47. B

    Complex integration over a square contour

    Homework Statement Let \Gamma be the square whose sides have length 5, are parallel to the real and imaginary axis, and the center of the square is i. Compute the integral of the following function over \Gamma in the counter-clockwise direction. You must use two different methods to solve...
  48. B

    Complex integration of real-valued trig function

    Homework Statement Integrate: \int \frac{1}{(3+2cos(θ))} dθ evaluated from zero to pi. Homework Equations I can't think of any. All of the integration formulas in the text rely on the existence of a singularity somewhere in the complex plane. This thing is analytic everywhere...
  49. D

    Complex Integration: Find g(2)=8πi, g(z) when |z|>3

    Homework Statement Let C be the circle |z|=3, described in the positive sense. Show that if g(z)= \int_C \frac{2s^2-s-2}{s-z} ds such that |z| does not equal 3, then g(2)=8 \pi i . What is the value of g(z) when when |z|>3? Homework Equations Cauchy Integral Formula Deformation of...
  50. H

    Complex Analysis Complex Integration Question

    Its question 1(g) in the picture. My work is shown there as well. This has to do with independence of path of a contour. Reason I am suspicious is that first there is a different answer online and second it says "principal branch" which I have not understood. Does that mean a straight line for...
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