1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    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!

Contour Integration with Legendre Functions

  1. Feb 24, 2007 #1
    1. The problem statement, all variables and given/known data
    P_n (z) and Q_n (z) are Legendre functions of the first and second kinds, respectively. The function f is a polynomial in z. Show that
    Q_n (z) = \frac{1}{2} P_n (z) \ln \left(\frac{z+1}{z-1} \right) + f_{n-1} (z)
    Q_n (z) = \frac{1}{2} \int \frac{P_n (t) \, dt}{z - t} \quad (n \text{ integer})
    by an application of Cauchy's formula. Be sure to specify the contour.

    [Hint: Q_n is many-valued. Cut the z-plane between -1 and 1 along the real axis.]

    2. Relevant equations
    Q_n (z) takes the form Q_n (x) in the region -1 < x < 1; x is real:
    Q_n (x) = \frac{1}{2} [Q_n (x + i\epsilon) + Q_n (x - i\epsilon)] =
    \frac{1}{2} P_n (x) \ln \left( \frac{1+x}{1-x} \right) + f_{n -1}(x) \, .

    Here's Cauchy's formula:
    f(z) = \frac{1}{2 \pi i} \oint \frac{f(t) \, dt}{t - z} \, .
    3. The attempt at a solution

    I tried choosing a barbell-shaped contour:


    The "O"s surround the points -1 and +1; the "=====" are a small distance away from the branch cut (x+ie for the upper part, and x-ie for the lower part, where e << 1). The integral over f is zero, since f is entire (it's a polynomial). Got lost after that.
    Last edited: Feb 24, 2007
  2. jcsd
  3. Feb 27, 2007 #2
    (Crickets chirping)
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Contour Integration Legendre Date
Contour Integration Jan 2, 2018
Contour Integration: Branch cuts Oct 28, 2017
Contour Integrals: Working Check Oct 21, 2017
Contour integration with a branch cut Feb 28, 2017
Contour integral using residue theorem Dec 4, 2016