(adsbygoogle = window.adsbygoogle || []).push({}); 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

[tex]

Q_n (z) = \frac{1}{2} P_n (z) \ln \left(\frac{z+1}{z-1} \right) + f_{n-1} (z)

[/tex]

implies

[tex]

Q_n (z) = \frac{1}{2} \int \frac{P_n (t) \, dt}{z - t} \quad (n \text{ integer})

[/tex]

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:

[tex]

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) \, .

[/tex]

Here's Cauchy's formula:

[tex]

f(z) = \frac{1}{2 \pi i} \oint \frac{f(t) \, dt}{t - z} \, .

[/tex]

3. The attempt at a solution

I tried choosing a barbell-shaped contour:

O======O

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Contour Integration with Legendre Functions

**Physics Forums | Science Articles, Homework Help, Discussion**