PHYS 598 hw3 - 1 Test functions and distributions Read the...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 498/MMA Handout 3 Fall 2007 Mathematical Methods in Physics I http://w3.physics.uiuc.edu/ m-stone5 Prof. M. Stone 2117 ESB University of Illinois 1) Test functions and distributions : Read the sections on distributions in chapter two of the lecture notes, then do the following problems: a) Let f ( x ) be a smooth function. i) Show that f ( x ) δ ( x ) = f (0) δ ( x ). Deduce that d dx [ f ( x ) δ ( x )] = f (0) δ 0 ( x ) . ii) We might also have used the product rule to conclude that d dx [ f ( x ) δ ( x )] = f 0 ( x ) δ ( x ) + f ( x ) δ 0 ( x ) . By integrating both against a test function, show this expression for the derivative of f ( x ) δ ( x ) is equivalent to that in part i). b) Let ϕ ( x ) be a test function. Using the definition of the principal part integrals , show that ∂t P Z -∞ ϕ ( x ) ( x - t ) dx = P Z -∞ ϕ ( x ) - ϕ ( t ) ( x - t ) 2 dx in two different ways: i) Fix the value of the cutoff . Differentiate the resulting -regulated integral, taking care to include the terms arising from the t dependence of the limits at x = t ± .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
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