Contour integration - Contour integration From Physics...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Contour integration From Physics Notes This article introduces contour integration , a powerful technique that will allow us to easily perform certain integrals that are otherwise difficult or impossible. Apart from their mathematical utility, contour integrals have rich physical consequences in many areas of physics, particularly in the study of waves and oscillations, as we'll see when discussing Fourier transforms and Green's functions. Contents 1 Contour integrals 1.1 Contour integral along a parametric curve 1.2 Example of a contour integral 2 Cauchy's integral theorem 2.1 Proof 2.2 Consequences 3 Poles 3.1 Residue theorem 3.2 Example of a loop integral 4 Using contours to solve integrals 4.1 Jordan's lemma 4.2 Another example 4.3 A more complicated example 5 Further reading 6 Exercises Contour integrals We have previously studied what it means to take the integral of a real function. By analogy, one can think of the integral of a complex function as the sum of at various closely-spaced points, multiplied by the infinitesimal displacements between each point. However, for complex numbers, the numbers being integrated over do not necessarily lie along a single dimension. Rather, they can be represented by points in a two dimensional complex plane. As shown in Fig. 1, we can imagine chaining together a sequence of points , which are separated by displacements , such that and so forth. Then the sum we are interested in is
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Fig. 1: Chaining a sequence of complex displacements to form an integral. As the displacements become infinitesimal, the sequence turns into a continuous curve (dark gray dotted curve), and the sum becomes a contour integral. In the limit where each displacement becomes infinitesimal (and the number of them becomes infinite), the sequence of points forms a continuous trajectory in the complex plane, which we denote by an abstract symbol . Then the above discrete sum turns into a contour integral (also called a line integral ), and is denoted like this: The symbol in the subscript of the integral sign indicates that the integral has to take place over the specific trajectory . It is always necessary, when talking about a contour integral, to specify the trajectory or "contour" that you're integrating over. This is similar to how, for a real definite integral, you have to state the endpoints of the integration range, except that in the complex case we specify an entire trajectory rather than just two numbers. It is also important to note that the contour implies a direction. If we integrate along the same curve but in the opposite direction, the value of the contour integral switches sign. Contour integral along a parametric curve In order to explicitly evaluate a contour integral along a curve , it is often useful to parameterize the curve. This is done by relating to some complex function of a real input. (As previously discussed, a complex function of a real variable describes a trajectory in the complex plane.) Let us denote this complex function of a real input by , where . The contour
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern