1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Using the method of steepest descent

  1. Oct 29, 2005 #1
    I have the question,
    [tex] \int_{-\pi/2}^{\frac{\pi}{2}} e^{-ilk}cos^n kdk[/tex]
    It says, "Set t=ik". So,
    [tex]-i\int_{-i\pi/2}^{i\pi/2}e^{-lt} cosh^n tdt[/tex]
    But then it says, "Use the method of steepest descent to show that as n [tex]\rightarrow \infty[/tex] with r = l/n."
    I'm supposed to get:
    [tex]\sim \sqrt{\frac{2\pi}{n(1-r^2)} }exp(-\frac{1}{2}n[r\log{\frac{1+r}{1-r}}+log(1-r^2)])[/tex]
    If the equation were of the form, [tex]\int e^{ilP(t)}Q(t)dt[/tex], I know how to use the method of steepest descent. I'd find a point z where P'(t)=0 and expand P(t) around that point using a Taylor series expansion getting, P(t)=P(z)+0.5P''(z)(t-z)^2, and then I'd replace t with z+ix and it would all come out from there. But I have no idea how to use the method of steepest descent when P(t)=t and i has been removed from the exponential.
    Last edited: Oct 29, 2005
  2. jcsd
  3. Oct 29, 2005 #2


    User Avatar
    Science Advisor
    Homework Helper

    HINT: Write [itex]\cosh ^n x = e^{\ln \cosh^n x}[/itex]
  4. Oct 29, 2005 #3
    :smile: Thanks, Tide. That question was killing me.
  5. Jul 20, 2011 #4
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?