Narrowband frequency modulation

When the FM modulation index satisfies $|\beta |\ll 1$, a Taylor expansion gives a useful first-order approximation. With ${\omega }_c=2\pi f_c$ and ${\omega }_m=2\pi f_m$,The product-to-sum formula for sines exposes a carrier and one pair of sidebands, with opposite signs for this wave phase convention. Higher sidebands are neglected, not identically absent. For $\beta =0.1$, the predicted normalized first-sideband magnitude $0.05$ is close to the original exact evaluation $J_1(0.1)\approx 0.0499375$. The neglected waveform term is bounded in absolute value by ${\beta }^2/2=0.005$, using the second-derivative bound for the cosine. Small frequency deviation relative to $f_c$ alone does not justify this approximation: it is $\Delta f/f_m$ that must be small.