Does anyone know how to show that $$ \int_{0}^{2\pi}H_{0}^{(1)}\left(2sin\left(\theta/2\right)x\right)e^{in\theta}d\theta=2\pi J_{n}(x)H_{n}^{(1)}(x) $$ where $n$ is an integer and $x>0$? It arose from solving a problem using two different methods. The results are only consistent when the integral above holds. Also, the following plots in Mathematica would seem to suggest it is correct:
Plot[Evaluate[Table[Re[BesselJ[n, x] HankelH1[n, x]], {n, 0, 3}]], {x,
0, 10}, PlotLegends -> Automatic]
Plot[Evaluate[Table[Im[BesselJ[n, x] HankelH1[n, x]], {n, 0, 3}]], {x,
0, 10}, PlotLegends -> Automatic]
ListPlot[Table[
Table[{x,
Re[NIntegrate[
HankelH1[0, 2 Sin[t/2] x] Exp[I n t], {t, 0,
2 \[Pi]}]/(2 \[Pi])]}, {x, 0, 10, 0.1}], {n, 0, 3}],
Joined -> True]
ListPlot[Table[
Table[{x,
Im[NIntegrate[
HankelH1[0, 2 Sin[t/2] x] Exp[I n t], {t, 0,
2 \[Pi]}]/(2 \[Pi])]}, {x, 0, 10, 0.1}], {n, 0, 3}],
Joined -> True]
I haven't been able to make progress so far. I was thinking a Fourier series representation of the Hankel function might be useful, but I end up going round in circles. There are some existing integral representations for Bessel functions, but I don't see how to make use of them here. Thanks in advance for any help.