Bessel Functions of the First and Second Kind

Bessel functions remain load-bearing across physics and engineering. In RF engineering, cylindrical-cavity modes and waveguide cutoffs are tabulated as zeros of $J_m$ and $J_m'$ — the TE$_{11}$ cutoff of a $2\ \mathrm{cm}$-diameter guide sits at the first zero of $J_1'=1.8412$, giving $f_c=8.79\ \mathrm{GHz}$. In photonics, the guided modes of a step-index fibre are $J_m$ in the core matched to modified functions $K_m$ in the cladding, and the single-mode cutoff is the zero $\alpha_{01}=2.405$ of $J_0$ — the number that tunes a kettledrum now sets whether a fibre carries one mode or many. FM synthesis encodes a signal in the Jacobi–Anger sidebands $J_n(\beta)$, so the "Bessel nulls" (carrier vanishing when $J_0(\beta)=0$ at $\beta=2.405$) calibrate modulation depth. And the Airy pattern governs the diffraction limit of every telescope and microscope — the James Webb Space Telescope's $6.5\ \mathrm{m}$ mirror uses the same $2J_1(u)/u$ point-spread function scaled up. An open computational thread remains: fast, uniformly accurate evaluation of high-order Bessel functions near the turning point $x\approx\nu$, where series and asymptotics both fail and uniform Airy-type expansions must take over.

Picture a cosine that has to live on a plane instead of a line. On the line a crest keeps its height forever; on the plane a cylindrical wavefront of fixed energy must spread over a circumference $2\pi\rho$ that grows with radius, so the amplitude fades. Bessel's $J_\nu(x)$ is exactly this compromise: it oscillates like a cosine but its envelope decays as $1/\sqrt{x}$, because intensity $\propto$ amplitude$^2$ must fall as $1/\rho$ to conserve the flux crossing each ring. That single geometric fact — energy diluted around a growing circle — predicts both the ringing and the slow decay of the large-argument form, and is the honest content of "cylindrical wave." Near the axis a different term rules. The $-\nu^2/x^2$ piece acts like a centrifugal barrier, forcing the solution to approach zero as $x^{\nu}$ (the regular $J_\nu$) or blow up as $x^{-\nu}$ (the singular content of $Y_\nu$). The order $\nu$ is the angular index $m$ from the separated factor $e^{im\phi}$, so more angular momentum makes the regular solution start flatter — a drum's higher-$m$ modes push their action toward the rim. The first figure shows the three canonical functions and the zeros that do all the quantizing.

The zeros are the workhorses. They are not perfectly periodic near the origin, but far out they settle toward a uniform gap of $\pi$, the residue of the buried cosine, with $\alpha_{\nu n}\to(n+\tfrac{\nu}{2}-\tfrac14)\pi$. Each zero is a node, and a Dirichlet condition on a circular boundary selects a discrete ladder of them, quantizing the radial wavenumber as the fixed ends of a string quantize its harmonics. But the drum's ladder is inharmonic — the zeros of $J_m$ are irrational multiples of one another, which is why a struck membrane sounds like a thud rather than a note. The second figure turns those zeros into the overtone spectrum they generate.

Finally, $Y_\nu$ is the "second polarization." Where $J_\nu$ asymptotes to a cosine, $Y_\nu$ asymptotes to a sine with the identical $1/\sqrt{x}$ envelope, lagging by a quarter cycle. Their Wronskian being nonzero everywhere is the cylindrical echo of $\cos$ and $\sin$ being linearly independent — and it lets any bounded-plus-singular radial field be written uniquely in terms of the pair.

Everything begins with the equation the radial separation leaves behind — a linear second-order ODE with a regular singular point at the origin, which is why a Frobenius series, not a Taylor series, is the right tool.

where: $x$ — dimensionless radial variable, usually $x=k\rho$ with $k$ a wavenumber; $\nu$ — the order, equal to the angular index $|m|$ in separated problems; $y(x)$ — the radial solution, a combination of $J_\nu$ and $Y_\nu$. Reading it: $x^2y''+xy'$ is the radial Laplacian, $+x^2y$ the wave term, and $-\nu^2y$ the centrifugal barrier from angular momentum. The regular solution is the Frobenius series built on the exponent $+\nu$, with reciprocal Gamma functions for coefficients — which is why the Gamma function is a prerequisite and why $J_\nu$ is entire in $x$.

where: $\Gamma$ — the Gamma function generalizing the factorial; $k$ — summation index. Reading it: the leading term $(x/2)^\nu/\Gamma(\nu+1)$ makes $J_\nu$ start as $x^\nu$, finite at the origin; the alternating sign turns the series into an oscillation once $x$ is large. The second, generally singular, solution is defined as a specific combination of $J_{\nu}$ and $J_{-\nu}$ chosen so that it is independent of $J_\nu$ for every order.

where: $Y_\nu$ — the Bessel function of the second kind, also the Neumann function $N_\nu$. Reading it: for non-integer $\nu$ this is a rotation within the space of $J_{\pm\nu}$; at integer $\nu=n$ both $\sin(\nu\pi)$ and the numerator vanish, and the limiting ratio injects the logarithm that makes $Y_n$ genuinely new. The certificate of independence is the Wronskian, and it has a famously simple closed form.

where: $W$ — the Wronskian of the pair, nonzero for every $x\in(0,\infty)$ and every $\nu$. Reading it: two solutions of a linear second-order ODE are independent iff their Wronskian is nonzero, so $2/(\pi x)\ne 0$ guarantees $J_\nu,Y_\nu$ span the full solution space — the most-used consistency check in the subject. Two representations tie the integer-order functions to plane waves: the generating function packs all $J_n$ into one exponential, and expanding a plane wave in cylindrical harmonics (Jacobi–Anger) follows immediately.

where: $t$ — an auxiliary complex variable whose $t^n$ coefficient is $J_n(x)$; $\theta$ — the angle between wavevector and position. Reading it: setting $t=e^{i\theta}$ turns the left identity into the right — the workhorse of cylindrical scattering theory, re-expressing a plane wave as partial waves of definite angular momentum. Neighbouring orders are linked by ladder-like recurrences, valid for $J_\nu$, $Y_\nu$, and the Hankel functions alike.

where: $\nu\pm1$ — the neighbouring orders. Reading it: the first relation climbs the order ladder, the second gives the derivative; at a zero of $J_\nu$ it yields $J_\nu'(\alpha_{\nu m})=-J_{\nu+1}(\alpha_{\nu m})$, the identity that fixes the Fourier–Bessel norm. On a finite disc the zeros of $J_\nu$ generate a complete orthogonal set — the Fourier–Bessel basis, the special-function analogue of an ordinary Fourier series with $\rho$ as the Sturm–Liouville weight.

where: $\alpha_{\nu m}$ — the $m$-th positive zero of $J_\nu$; $a$ — the disc radius; $\rho$ — radial coordinate and weight; $\delta_{mn}$ — Kronecker delta. Reading it: distinct radial modes are orthogonal, and the diagonal norm $\tfrac{a^2}{2}J_{\nu+1}^2(\alpha_{\nu m})$ is what you divide by when projecting data onto the modes.

:::derivation[The Wronskian W[Jν, Yν] = 2/(πx) and why the pair always spans the solution space] The goal is to prove that $J_\nu$ and $Y_\nu$ are linearly independent for every order and argument by computing their Wronskian in closed form, showing its value is forced by the structure of the equation rather than the detailed series. Abel's identity fixes the $x$-dependence directly from the coefficient of $y'$, leaving one multiplicative constant undetermined; that constant is read off from the leading behaviours near the origin. The payoff is the certificate at the heart of the key idea: a Wronskian manifestly nonzero on all of $(0,\infty)$.

Step 1. In standard form $y''+p(x)y'+q(x)y=0$ the coefficient of $y'$ is $p=1/x$. Why: Abel's theorem gives $W(x)=W(x_0)\exp[-\int p\,\mathrm dx]$, so only $p$ — not the complicated $q$ — controls the Wronskian, and $\int(1/x)\,\mathrm dx=\ln x$.

Step 2. Abel's theorem then gives $W(x)=C\,e^{-\ln x}=C/x$. Why: whatever the solutions are, $W$ can only be $C/x$; this already shows it cannot vanish at finite $x>0$ unless identically zero, reducing the independence question to one number.

Step 3. Insert the small-$x$ forms $J_\nu\sim\tfrac{1}{\Gamma(\nu+1)}(x/2)^\nu$ and $Y_\nu\sim-\tfrac{\Gamma(\nu)}{\pi}(2/x)^\nu$ into $W=J_\nu Y_\nu'-J_\nu'Y_\nu$. Why: since $W=C/x$ exactly, evaluating it in any limit gives $C$ for all $x$; the origin is convenient because both functions reduce to single powers there.

Step 4. The two products combine, using $\Gamma(\nu+1)=\nu\Gamma(\nu)$, into $W=\tfrac{1}{x}\cdot\tfrac{2\nu}{\pi}\cdot\tfrac{1}{\nu}=\dfrac{2}{\pi x}$, so $C=2/\pi$ and $W\ne0$ everywhere. Why: a nonvanishing Wronskian is necessary and sufficient for linear independence, so this one line certifies that $\{J_\nu,Y_\nu\}$ is a basis at every order — the guarantee the whole theory rests on. *As a numerical check: $J_0(x)Y_0'(x)-J_0'(x)Y_0(x)=2/(\pi x)$ holds at $x=1,3,10$ to five digits (e.g. $0.6366/x$), and the $x$-dependence is exactly the $1/x$ form that Abel's theorem requires.*

The two extremes plus the integer-order degeneracy account for nearly every physical use of these functions.

1 · The overtone spectrum of a circular drum (fully worked)

A clamped circular membrane of radius $a=0.15\ \mathrm{m}$ carries transverse waves at speed $c=100\ \mathrm{m/s}$. Find its fundamental frequency and the first few overtones, and show they are inharmonic.

Step 1. Separation gives modes $u\propto J_m(k\rho)\,e^{im\phi}\,e^{-i\omega t}$; regularity on the axis forces $J_m$ (not $Y_m$), and the clamped edge $u(a)=0$ demands $J_m(ka)=0$, so $ka=\alpha_{mn}$. Hence $\omega_{mn}=c\,\alpha_{mn}/a$ and $f_{mn}=c\,\alpha_{mn}/(2\pi a)$.

Step 2. The zeros are $\alpha_{01}=2.4048$, $\alpha_{11}=3.8317$, $\alpha_{21}=5.1356$, $\alpha_{02}=5.5201$. With $c/(2\pi a)=100/(2\pi\cdot0.15)=106.10\ \mathrm{s^{-1}}$, the fundamental is $f_{01}=106.10\times2.4048=\mathbf{255.2\ Hz}$.

Step 3. The next modes share the multiplier: $f_{11}=\mathbf{406.5\ Hz}$, $f_{21}=\mathbf{544.9\ Hz}$, $f_{02}=\mathbf{585.7\ Hz}$.

Step 4. Normalized to the fundamental, the ratios are $1.000,\ 1.593,\ 2.136,\ 2.295$. These are not integers — unlike a string's $1,2,3,\dots$, the drum's overtones are irrational multiples set by the zeros of $J_m$, the mathematical reason a struck membrane produces a pitchless thud. Fundamental 255.2 Hz; overtone ratios 1.000, 1.593, 2.136, 2.295 — inharmonic.

2 · The Airy disc and the Rayleigh resolution limit

Fraunhofer diffraction through a circular aperture of diameter $D$ gives a far-field amplitude $\propto 2J_1(u)/u$ with $u=\tfrac12 kD\sin\theta$. Locate the first dark ring and evaluate the angular resolution of the $2.4\ \mathrm{m}$ Hubble mirror at $\lambda=550\ \mathrm{nm}$.

Step 1. The pattern is dark where $J_1(u)=0$ with $u\ne0$, i.e. at the first positive zero $u=\alpha_{11}=3.8317$ (the central maximum sits at $u=0$, where $2J_1(u)/u\to1$). So the first dark ring is at $\tfrac12 kD\sin\theta=3.8317$.

Step 2. With $k=2\pi/\lambda$, solve for the angle: $\sin\theta=\dfrac{2\times3.8317}{kD}=\dfrac{3.8317\,\lambda}{\pi D}=\dfrac{3.8317}{\pi}\cdot\dfrac{\lambda}{D}$. The numerical prefactor is $3.8317/\pi=\mathbf{1.2197}$, which is precisely where the famous Rayleigh constant $1.22$ comes from — it is a zero of $J_1$ divided by $\pi$, not an empirical fudge.

Step 3. Thus $\theta_{\min}\approx1.22\,\lambda/D=1.2197\times(550\times10^{-9})/2.4=\mathbf{2.80\times10^{-7}\ rad}$, or $\times206265=\mathbf{0.058\ arcsec}$.

Step 4. Interpretation: two stars closer than $0.058''$ blur into one for Hubble at this wavelength — the diffraction limit, fixed entirely by the first zero of $J_1$. First dark ring at $J_1=0$ (u = 3.8317); Rayleigh factor 1.22 = 3.8317/π; Hubble resolution ≈ 0.058″.

The functions carry the name of the astronomer who first studied them systematically, but their story runs from a swinging chain to the modern language of scattering.

Texts. Arfken, Weber & Harris, Mathematical Methods for Physicists, 7th ed., Ch. 14 (Bessel Functions) · Hassani, Mathematical Physics, Ch. on Second-Order ODEs and Sturm–Liouville systems · Stone & Goldbart, Mathematics for Physics, Ch. 8–9 (special functions, Green functions) · Byron & Fuller, Mathematics of Classical and Quantum Physics, Ch. 6 · Abramowitz & Stegun, Handbook of Mathematical Functions, Ch. 9 · Watson, A Treatise on the Theory of Bessel Functions (the definitive reference). Key results used throughout: indicial roots $\pm\nu$ and recursion $a_k=-a_{k-2}/[k(k+2\nu)]$; $J_{-n}=(-1)^nJ_n$ via the Gamma-pole argument; $W=2/(\pi x)$ from Abel's theorem plus small-$x$ forms; asymptotic envelope $\sqrt{2/\pi x}$ with phase $\nu\pi/2+\pi/4$; Fourier–Bessel norm $\tfrac{a^2}{2}J_{\nu+1}^2(\alpha_{\nu m})$. Numerical anchors: zeros $\alpha_{01}=2.4048$, $\alpha_{02}=5.5201$, $\alpha_{11}=3.8317$, $\alpha_{21}=5.1356$; first zero of $J_1'=1.8412$; $J_{1/2}(3)=0.06501$; drum ratios $1.593,\,2.136,\,2.295$; Rayleigh $3.8317/\pi=1.220$; $\gamma=0.5772$.

Prerequisites. The Gamma Function · Analytic Continuation and Poles of the Gamma Function · Orthogonal Polynomials: General Theory. Leads to. Modified Bessel Functions · Spherical Bessel Functions · Hankel Functions and Asymptotic Behaviour · Recurrence Relations for Special Functions · Generating Functions for Special Functions.

Ask the tutor: Ask me to walk through why the Wronskian $2/(\pi x)$ falls out of Abel's theorem, or to expand any cylindrical field of yours in a Fourier–Bessel series.