Integral with legendre generating function

[buffordboy23]

Homework Statement

Use the Legendre generating function to show that for A > 1,

$$\int^{\pi}_{0} \frac{\left(Acos\theta + 1\right)sin\thetad\theta}{\left(A^{2}+2Acos\theta+1\right)^{1/2}} = \frac{4}{3A}$$

Homework Equations

The Legendre generating function

$$\phi\left(-cos\theta,A\right) = \left(A^{2}+2Acos\theta+1\right)^{-1/2}} = \sum^{\infty}_{n=0}P_{n}\left(-cos\theta\right)A^{n}$$

The Attempt at a Solution

Pretty much clueless, and the book makes no mention of anything else. I know that with these parameters the series should be convergent for |A| < 1, and that the interval now runs from [-pi, pi]. The only thing on the internet that I could find that deals with with integrals of Legendre functions is for orthogonality, which doesn't seem to apply here.

 

[Pere Callahan]

Homework Statement

Use the Legendre generating function to show that for A > 1,

$$\int^{\pi}_{0} \frac{\left(Acos\theta + 1\right)sin\thetad\theta}{\left(A^{2}+2Acos\theta+1\right)^{1/2}} = \frac{4}{3A}$$
.

Using the trick the integral becomes

$$/int_0^\pi{\left(A\cos\theta+1\right)\sin\theta\sum_{n=0}^\infty{P_n(-\cos\theta)A^n}}$$

I suggest you are courageous and interchange the integration and summation (or maybe you can argue that this is allowed in our case?) then you expand $A\cos\theta +1$ in terms of Legendre functions. You should then be able to use some orthoganality property.

 

[buffordboy23]

I'm still stuck with this problem. Here is my work.

Let $$x = -cos\theta \rightarrow d\theta = dx/sin\theta$$. Substitution gives

$$\int^{1}_{-1} \frac{\left(1-Ax\right)dx}{\left(A^{2}-2Ax+1\right)^{1/2}} = \int^{1}_{-1} \left(\sum^{\infty}_{k=0}P_{k}\left(x\right)A^{k}\right)\left[1-Ax\right]dx = \int^{1}_{-1} \left(\sum^{\infty}_{k=0}P_{k}\left(x\right)A^{k}\right)\left[P_{0}\left(x\right)-P_{1}\left(x\right)A\right]dx = \int^{1}_{-1} \left(P_{0}\left(x\right)^{2}-P_{1}\left(x\right)^{2}A^{2}\right)dx$$

where the last equation exploited orthogonality. Yet,

$$\int^{1}_{-1} \left(P_{0}\left(x\right)^{2}-P_{1}\left(x\right)^{2}A^{2}\right)dx = 2 - \frac{2A^{2}}{3}$$

and is not the answer. What am I missing?

I know that the legendre generating function requires |A|<1, but this problem says A>1. This implies that the series is divergent, but does that necessarily matter in this context?

 

[gabbagabbahey]

Use the Legendre generating function to show that for A > 1

The Legendre generating function

$$\phi\left(-cos\theta,A\right) = \left(A^{2}+2Acos\theta+1\right)^{-1/2}} = \sum^{\infty}_{n=0}P_{n}\left(-cos\theta\right)A^{n}$$

... I know that with these parameters the series should be convergent for |A| < 1

For A>1, this is not the generating function.

Hint: if A>1, then 1/A<1

 

[buffordboy23]

Here we go.

$$\int^{1}_{-1} \frac{\left(1-Ax\right)dx}{\left(A^{2}-2Ax+1\right)^{1/2}} = \int^{1}_{-1} \frac{\left(1/A-x\right)dx}{\left(1-2x/A+1/A^{2}\right)^{1/2}} =\int^{1}_{-1} \left(\sum^{\infty}_{k=0}P_{k}\left(x\right)A^{-k}\right)\left[1/A-x\right]dx = \int^{1}_{-1} \left(\sum^{\infty}_{k=0}P_{k}\left(x\right)A^{-k}\right)\left[P_{0}\left(x\right)/A-P_{1}\left(x\right)\right]dx$$

$$= \int^{1}_{-1} \left(\frac{P_{0}\left(x\right)^{2}}{A}-\frac{P_{1}\left(x\right)^{2}}{A}\right)dx = \frac{1}{A}\int^{1}_{-1} \left(1-x^{2}\right)dx = \frac{4}{3A}$$

Thanks.