What is Complex integral: Definition and 125 Discussions

In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as




{\displaystyle W=\mathbf {F} \cdot \mathbf {s} }
, have natural continuous analogues in terms of line integrals, in this case









{\displaystyle \textstyle W=\int _{L}\mathbf {F} (\mathbf {s} )\cdot d\mathbf {s} }
, which computes the work done on an object moving through an electric or gravitational field F along a path


{\displaystyle L}

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  1. G

    Complex contour integral proof

    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?
  2. H

    Evaluating cosine function from ##-\infty## to ##\infty##

    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...
  3. Antarres

    A A question about a complex integral

    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...
  4. tworitdash

    A Integrating a function of which poles appear on the branch cut

    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...
  5. Robin04

    Calculating a complex integral

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

    How Do You Solve a Complex Integral Using Cauchy-Goursat's Theorem?

    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...
  8. S

    A Derivation of a complex integral with real part

    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...
  9. Rectifier

    Complex logarithm as primitive

    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...
  10. W

    Numerical/Analytical Solution to a Complex Integral

    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##...
  11. W

    Complex Integral to error function

    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...
  12. S

    I Solving Complex Integral Paths - Real Line Poles

    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...
  13. S

    I Prove Complex Integral: $\int_m^\infty\sqrt{x^2-m^2}e^{-ixt}dx$

    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?
  14. S

    I Complex integral of a real integrand

    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...
  15. J

    I Can the Complex Integral Problem Be Solved Using Residue Theorem?

    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...
  16. Leechie

    Evaluating complex integral problem

    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}...
  17. B

    MHB How to Evaluate the Integral of z(z+1)cosh(1/z) Over a Unit Circle?

    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...
  18. Enrike

    Complex integral in finite contour at semiaxis

    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...
  19. ognik

    MHB Please check this complex integral (#2)

    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} =...
  20. ognik

    MHB Please check this complex integral

    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})=...
  21. D

    Real integral=area , complex integral= ?

    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 ?
  22. T

    Complex integral for z-transform causality

    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...
  23. J

    Solve Complex Integral: Find Residues & Singularities

    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...
  24. ShayanJ

    How can the difficulty of a complex integral be solved?

    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!
  25. K

    MHB Absolute Value of Complex Integral

    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...
  26. K

    2nd order pole while computing residue in a complex 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}...
  27. KleZMeR

    Complex Integral Trigonometric Substitution

    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?
  28. C

    Remind me how to do this complex integral

    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?
  29. L

    MHB Solving a Complex Integral: Substituting tg (x/2)

    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)
  30. M

    Why Does the Integral of e^(iz)/z Over an Expanding Semicircle Converge to Zero?

    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}...
  31. S

    MHB Evaluating the Complex Integral

    \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...
  32. J

    Calculating a real integral with a complex integral

    [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...
  33. A

    Solve Complex Integral: Residual Calculus?

    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
  34. J

    Understanding Complex Integrals: Interpretation and Visualization

    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!
  35. B

    Proving the Value of a Complex Integral Involving Cosecant and the Unit Circle

    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...
  36. E

    Calculate complex integral as line integral

    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?
  37. P

    Complex integral is zero but fn. is not analytic

    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 )...
  38. B

    The Limit of a Complex Integral

    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##...
  39. J

    Heat conduction (Fourier complex integral)

    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...
  40. A

    Complex integral coming from a 1loop diagram

    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...
  41. R

    How Do You Evaluate Complex Contour Integrals with Sinusoidal Functions?

    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...
  42. B

    Evaluate this complex Integral

    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...
  43. R

    Calculating Antiderivative of Complex Integral

    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...
  44. U

    Complex integral over a circle

    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...
  45. T

    Parametric definition for a complex integral

    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...
  46. H

    MHB Is My Complex Integral Calculation Correct?

    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 $$...
  47. A

    Contour Integration and Residues: Solving Complex Integrals for Summation

    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) -...
  48. R

    Evalution of a complex integral

    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...
  49. polygamma

    MHB Is this approach to evaluating a complex integral valid?

    $\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}...
  50. D

    MHB Complex Integral: $\int_0^1$ Calculation

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