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



W
=

F

⋅

s



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




W
=

∫

L



F

(

s

)
⋅
d

s




{\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



L


{\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

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  3. Antarres

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    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

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  5. Robin04

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  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

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  8. S

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  9. Rectifier

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  10. W

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  11. W

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  12. S

    I Solving Complex Integral Paths - Real Line Poles

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  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

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  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

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  17. B

    MHB Evaluating a complex integral

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  18. Enrike

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  19. ognik

    MHB Please check this complex integral (#2)

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  20. ognik

    MHB Please check this complex integral

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  21. D

    Real integral=area , complex integral= ?

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  22. T

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  23. J

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  24. ShayanJ

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  25. K

    MHB Absolute Value of Complex Integral

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  26. K

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  27. KleZMeR

    Complex Integral Trigonometric Substitution

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  28. C

    Remind me how to do this complex integral

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  29. L

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

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  30. M

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  31. S

    MHB Evaluating the Complex Integral

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  32. J

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  33. A

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  34. J

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    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

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  36. E

    Calculate complex integral as line integral

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  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

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  39. J

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  40. A

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  41. R

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  42. B

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  43. R

    Calculating Antiderivative of Complex Integral

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  44. U

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  45. T

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  46. H

    MHB Evaluate a complex integral

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  47. A

    Contour Integration and Residues: Solving Complex Integrals for Summation

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  48. R

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  49. polygamma

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  50. D

    MHB Complex Integral: $\int_0^1$ Calculation

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