I'm trying to prove this integral representation:
$$J_\nu = \frac{({x/2})^\nu}{\Gamma(\nu+1/2)\sqrt{\pi}}\int_{-1}^{1}(1-t^2)^{\nu-1/2}e^{itx}dt$$
I think countour integration would be useful, but in this case I'm trying to avoid direct integration.
I've tried changing variables of the integral and used the generating function:
$$\int_{-1}^{1}(1-t^2)^{\nu-1/2}e^{itx}dt=\int_{-\pi/2}^{\pi/2}(\cos \theta)^{2\nu}e^{ix\sin\theta}d\theta=\int_{-\pi/2}^{\pi/2}(\cos \theta)^{2\nu}\left(\sum _{n=-\infty}^\infty J_ne^{in\theta}\right)d\theta$$
I think I need to use somewhere the Beta function.
I would prefer a hint, not the whole proof.
The typical way to derive it is to expand $e^{itx}$ in its Maclaurin series, switch the order of summation and integration, integrate, and then apply the duplication formula for the gamma function.
$$ \begin{align} \int_{-1}^{1}(1-t^2)^{\nu-1/2}e^{itx} \, dt &= \int_{-1}^{1} (1-t^{2})^{\nu-1/2} \sum_{n=0}^{\infty}\frac{(itx)^{n}}{n!} \, dt \\ &= \sum_{n=0}^{\infty} \frac{(ix)^{n}}{n!} \int_{-1}^{1} t^{n} (1-t^{2})^{\nu-1/2} \, dt \\ &= 2 \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \int_{0}^{1} t^{2n} (1-t^{2})^{\nu-1/2} \, dt \tag{1} \\ &= \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \int_{0}^{1} u^{n-1/2} (1-u)^{\nu-1/2} \, du \\ &= \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} B \left(n + \frac{1}{2}, \nu + \frac{1}{2} \right) \\ &=\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \frac{\Gamma(n+ \frac{1}{2}) \Gamma(\nu+\frac{1}{2})}{\Gamma(n+\nu+1)} \\ &= \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!} \frac{2^{1-2n} \sqrt{\pi} \ \Gamma(2n) \Gamma(\nu+\frac{1}{2})}{\Gamma(n) \Gamma(n+\nu+1)} \frac{n}{n} \tag{2} \\ &= \sqrt{\pi} \, \Gamma \left(\nu + \frac{1}{2} \right)\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n! \Gamma(n+\nu+1)} \left(\frac{x}{2} \right)^{2n} \\ &= \sqrt{\pi} \, \Gamma \left( \nu+\frac{1}{2} \right) \left( \frac{2}{x}\right)^{v} J_{\nu}(x)\end{align}$$
$(1)$ The integrand is even if $n$ is even and odd if $n$ is odd.
$(2)$ Duplication formula for the gamma function