Chapter 12.23, Problem 20MP

In the generating function equation of Problem 19, put $h=e^{i\theta }$ and separate real and imaginary parts to derive the equations $\cos (x\sin \theta )=J_0(x)+2J_2(x)\cos 2\theta +2J_4(x)\cos 4\theta +\cdots$

$=J_0(x)+2{\displaystyle \underset{n=1}{\sum }{J}_{2n}}(x)\cos 2n\theta$, $=2{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}{J}_{2n+1}}(x)\sin (2n+1)\theta$.

These are Fourier series with Bessel functions as coefficients. (In fact the $J_n$ ’s for integral $n$ are often called Bessel coefficients because they occur in many series like these.) Use the formulas for the coefficients in a Fourier series to find integrals representing $J_n$ for even $n$ and for odd n. Show that these results can be combined to give $J_n(x)=\frac{1}{\pi }{\displaystyle {\int }_0^{\pi }\cos }(n\theta -x\sin \theta )d\theta$ for all integral n. These series and integrals are of interest in astronomy and in the theory of frequency modulate waves.