Math Help - bessel function-legendre polynomials

1. bessel function-legendre polynomials

Can anyone help me to solve integral of function exp(i alfa x)P_n(x)dx
in limits form -1 to 1
where P_n(x) is legendre polynomial.
Solution sholud be
sqr(2pi/alfa)iJ_n+1/2(alfa)
where J_n+1/2(alfa)
is modifided bessel function
pls help!
and sorry that i written it like this but i forgot how to use latex

2. Re: bessel function-legendre polynomials

One way to prove this is to start with the spherical harmonic expansion of a plane wave. Alternatively, you could verify directly that $J_{n+1/2}$ satifies the approproaite d.e. with initial conditions.

3. Re: bessel function-legendre polynomials

can you please show me how to verify this.

4. Re: bessel function-legendre polynomials

What text are you using?

5. Re: bessel function-legendre polynomials

an introduction to the teory of functions of a complex variable by e.t.copson
page 341.

7. Re: bessel function-legendre polynomials

I'll take a look at your text tomorrow. In the meantime, here's the plane wave expansion

$e^{ikr\cos \gamma}=\sum_{n=0}^\infty a_nj_n(kr)P_n(\cos \gamma)$.

where $j_n$ is the spherical Bessel function. From this you can deduce the integral representation of $j_n$.

8. Re: bessel function-legendre polynomials

OK, I have his book and I located the question. It seems he wants to deduce Bauer's formula from the result and so we can't start with the plane wave expansion. Give me a little time to think of the argument intended by the author.

9. Re: bessel function-legendre polynomials

Here's the argument I think Copson is expecting. From example 2 on page 320 we have

$J_\nu (z)=\frac{(\frac{1}{2}z)^\nu}{\Gamma(\nu+\frac{1}{ 2})\Gamma(\frac{1}{2})}\int_0^\pi\cos(z\cos \theta)\sin^{2\nu}\theta\, d\theta$.

Now making the substitution $t=\cos \theta$ gives

$J_{n+\frac{1}{2}}=\frac{(\frac{1}{2}z)^{n+\frac{1} {2}}}{\Gamma(n+1)\Gamma(\frac{1}{2})}\int_{-1}^1\cos(zt)(1-t^2)^n\, dt$.

Now integrate $n$ times by parts and use Rodrigues's formula

$P_n(t)=\frac{1}{2^nn!}\frac{d^n}{dt^n}(t^2-1)^n$.

10. Re: bessel function-legendre polynomials

Thank you very much for your help!