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For a nonnegative integer $m$, expand the plane wave in powers of $\rho$:The first contribution to the positive angular harmonic $e^{im\theta }$ occurs at $n=m$. Its coefficient is $(i\rho {)}^m/(2^{m}m!)$. Comparing this with $a_m{J}_m(\rho )\sim a_m{\rho }^m/(2^{m}m!)$ gives $a_m=i^m$ for $m\ge 0$.

The plane wave is even in $\theta$, so its $m$ and $-m$ Fourier coefficients agree. Therefore $a_{-m}{J}_{-m}=a_m{J}_m$, implying $a_{-m}=(-1{)}^m{i}^m=i^{-m}$. HenceThis is the cosine form of the Jacobi-Anger expansion. The factorial asymptotic stated in the source is used for nonnegative order; applying it literally to negative integers would introduce an undefined factorial. The Integer-order Bessel reflection identity is the required extension, not a separate set of divergent radial solutions.

The incident plane wave is periodic in $\theta$, so expand it as a Fourier series ${\sum }_m{R}_m(r)e^{im\theta }$. Substitution into the Helmholtz equation, with $\rho =kr$, gives the radial Bessel differential equationRegularity at the origin selects the Bessel function of the first kind, rather than the singular second solution. ThusThe coefficients $a_m$ are independent of $r$; their values follow from matching near the origin in the next part. The completed identity is the Jacobi-Anger expansion.

For $m\ge 0$, expand $e^{i\rho \cos \theta }=\exp [i\rho (e^{i\theta }+e^{-i\theta })/2]$. The lowest power producing the Fourier harmonic $e^{im\theta }$ is $(i\rho /2{)}^m/m!$. Meanwhile $J_m(\rho )\sim (\rho /2{)}^m/m!$. Matching these lowest powers gives $a_m=i^m$.

For negative order use $J_{-m}=(-1{)}^m{J}_m$ and the equality of the positive and negative Fourier coefficients of this even function of $\theta$. This gives $a_{-m}=(-1{)}^m{i}^m=i^{-m}$. Therefore the Jacobi-Anger expansion isThe cosine form has multiplicity one for $m=0$ and two for every positive $m$. Those multiplicities are essential for matching the incident plane wave.

Let $w_0=1$ and $w_m=2$ for $m\ge 1$, as required by the cosine form of the Jacobi-Anger expansion. For a central real potential, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. Splitting each shifted radial cosine into incoming and outgoing exponentials leaves its incoming coefficient unchanged and multiplies its outgoing coefficient by $S_m$. Subtracting the incident plane wave givesComparing with the prescribed scattering amplitude convention yields the two-dimensional partial-wave scattering formula,Orthogonality gives ${\int }_0^{2\pi }1d\theta =2\pi$ and ${\int }_0^{2\pi }{\cos }^2(m\theta )d\theta =\pi$ for $m\ge 1$, while all cross terms vanish. HenceThe multiplicities printed in this part are reversed. With those printed values and every ${\delta }_m=0$, the purported free wave would be $\frac{1}{2}{e}^{ikr\cos \theta }+\frac{3}{2}{J}_0(kr)$, rather than the required incident plane wave. The extra $J_0$ contains an incoming cylindrical wave, which cannot be absorbed into $f$. The formulas above use the corrected multiplicities and therefore satisfy the specified scattering boundary condition.

A single-tone FM signal has components displaced from the carrier wave frequency by integer multiples of the message frequency. These are temporal sidebands; the spatial sidebands used to perturb a spatial pattern are a related but different setting. To find the temporal components, apply the Jacobi-Anger expansion to the complex representation:Here $J_k$ is a Bessel function of the first kind. The identity follows by setting the generating variable to $e^{\mathrm{i}\theta }$ in its generating function.^[ref]

Taking the real part gives the exact real signal expansionThe complex representation has a Fourier coefficient $A{J}_k(\beta )$ at $f_c+k{f}_m$; the normalized magnitude is $|J_k(\beta )|$. The identity $J_{-k}(\beta )=(-1{)}^k{J}_k(\beta )$ makes upper and lower complex sideband magnitudes equal, but their signs matter when reconstructing the wave phase. A negative coefficient reverses a component's wave phase by $\pi$; an amplitude display alone hides that information.

For $\beta =5$, original evaluation gives $J_0(5)\approx -0.17760$, $J_1(5)\approx -0.32758$, $J_5(5)\approx 0.26114$ and $J_6(5)\approx 0.13105$. Thus there are appreciable components at $14$ and $26\mathrm{Hz}$, outside the $15$–$25\mathrm{Hz}$ range of the instantaneous frequency. There is no contradiction: instantaneous frequency describes the local rate of the combined wave phase, while the frequency spectrum describes a global superposition of constant-frequency components.

For the real signal, negative frequencies must also be accounted for: a cosine at $-f$ equals one at $f$. When $f_c/f_m=20$, the normalized positive-frequency cosine coefficient at $n{f}_m$, $n>0$, is exactly $J_{n-20}(\beta )+J_{-n-20}(\beta )$. The second term is extremely small over the plotted range, but the figure includes it. The chosen complex representation need not have a frequency spectrum supported only at positive frequencies, even when its specified instantaneous frequency is everywhere positive.

For a nonzero FM modulation index, the ideal frequency spectrum has infinitely many sidebands. Many become tiny, but drawing only visible lines does not make the exact signal bandwidth finite. A useful engineering width must state a threshold or a retained-power fraction. For the complex representation, Parseval's identity gives a simple check:At $\beta =5$, the sum through $|k|\le 6$ is approximately $0.99356$. Retaining those complex components captures over $99\%$ of their squared magnitudes and spans $12\mathrm{Hz}$ from $14$ to $26\mathrm{Hz}$. This is a stated truncation criterion, not an exact cutoff or a universal bandwidth rule. It also explains why a diminished carrier component need not imply a diminished envelope: its contribution has been redistributed among sidebands.

For a real central potential and the outgoing convention $e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. The cosine form of the Jacobi-Anger expansion fixes $w_0=1$ and $w_m=2$ for $m\ge 1$. Comparing the outgoing part of each phase-shifted radial cosine with the incident plane wave gives the displayed scattering amplitude. Integrating $|f{|}^2/k$ gives the displayed scattering cross-section, since the constant angular mode has squared norm $2\pi$ and each positive-order cosine has squared norm $\pi$. Reversing the multiplicities would already fail to reproduce the free plane wave at zero scattering phase shift.