Calculate some integrals with Chebyshev polynomials and Bessel functions?

Questions
1

Hello,

Recently I want to evaluate some Integrals with Chebyshev polynomials and Bessel functions. I found the Mathematica came out with no analytical solutions (The results were the same as I typed in.) Therefore I tried

Integrate[Cos[n*theta]*E^(I*w*W*Cos[theta]), {theta, 0, Pi}, 
 Assumptions -> {Element[{W, theta}, Reals], Element[n, Integers]}]

And it still showed no analytical results. However, there is one Bessel identity, which specifies

\int_o^\pi \cos n\theta e^{jwW\cos\theta}d\theta=\pi j^n J_n (wW)

where J_n is the bessel functions of order n

Can Mathematica deal with integration with Bessel identities? Or I made any mistakes.

2

Possible duplicate.

Table[Integrate[Cos[n*theta]*E^(I*w*W*Cos[theta]), {theta, 0, Pi}, Assumptions -> {Element[{W, w, theta}, Reals]}], {n, 1, 
3}] // MatrixForm

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