# Contour integration exp(ikx)/x

• aranud
In summary: You get this:limε→0 RE[int(sin(kx)/(x+iε)]+i limε→0 RE[int(icos(kx)/(x+iε))] = 0This means that the two integrals are both zero, which is not what you want. You need to solve for k so that it is positive, and then you can do the displacement.
aranud

## Homework Statement

Calculate the principal value integral exp(ikx)/x from -infinity to infinity first with a formula derived in the textbook and then by displacing the pole. Use this result to calculate the integral of sin(x)/x from -infinity to infinity.

## Homework Equations

If the integral around the contour in the upper half of the plane goes to zero when the radius goes to infinity one can use the following formula:
integral f(x)/(x-x0) = i*pi/f(x0)+2pi*(residus poles) The poles in the domain around which you integrate.

## The Attempt at a Solution

I've managed to prove with jordan's lemma that the upper part of the contour goes to zero when integrated so the formula can be used. Since the only pole is on the contour at x=0 we get that the integral is i*pi*1. This isn't so weird because the integrand isn't real either.

My problem is with the displacement of the pole technique. If we displace the pole north by changing the integrand to exp(ikx)/(x+iε) the integrand is split in an imaginary and real part:
limε→0 RE[int(sin(kx)/(x+iε)] + i limε→0 RE[int(icos(kx)/(x+iε))]

But because there are no poles in the domain around which I integrate these should both be zero? I think I'm missing something very basic here but Ican't figure out what it is.

Thanks.

The way you close the contour should depend on the sign of k. Did you take that into account?

Well the problem statement just said k was real number so I tacitly assumed it to be positive and closed the contour in the half plane. The integral shouldn't depend on which half-plane you use to close the contour anyway.
edit: apparently the sign of the integral does depend on k. But if I manage to solve it for k positive I'll understand the k negative problem too.

aranud said:
Well the problem statement just said k was real number so I tacitly assumed it to be positive and closed the contour in the half plane. The integral shouldn't depend on which half-plane you use to close the contour anyway.
edit: apparently the sign of the integral does depend on k. But if I manage to solve it for k positive I'll understand the k negative problem too.

For the first calculation, you can just change your variables so that the exponential eikx is replaced with ei|k|x, and you get your result easily. But what happens if you try doing after you have displaced the pole?

## 1. What is the mathematical definition of "Contour integration exp(ikx)/x"?

Contour integration exp(ikx)/x is a mathematical technique used to evaluate integrals of the form ∫f(z) dz, where f(z) is a complex-valued function of a complex variable z. It involves integrating the function along a specified path or contour in the complex plane.

## 2. What is the importance of "Contour integration exp(ikx)/x" in mathematics?

Contour integration exp(ikx)/x is an essential tool in complex analysis, which has applications in various fields of mathematics, including number theory, differential equations, and physics. It allows for the evaluation of difficult integrals and the solution of boundary value problems.

## 3. How does "Contour integration exp(ikx)/x" differ from ordinary integration?

The main difference between contour integration exp(ikx)/x and ordinary integration is that the former is performed in the complex plane while the latter is performed in the real plane. In contour integration, the path of integration is specified, whereas in ordinary integration, the limits of integration are specified.

## 4. What are some common applications of "Contour integration exp(ikx)/x"?

Contour integration exp(ikx)/x has various applications in mathematics, physics, and engineering. It is used to solve differential equations, evaluate improper integrals, and find the residues of complex functions. It also has applications in signal processing, control theory, and probability theory.

## 5. Are there any limitations to using "Contour integration exp(ikx)/x"?

One limitation of contour integration exp(ikx)/x is that it can only be used to evaluate integrals of functions that are analytic in the complex plane. This means that the function must be differentiable at all points in the contour. In addition, finding the correct contour and calculating the residues can be challenging for complex functions with many singularities.

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