Bessel Functions of the First and Second Kind

Bessel functions

J_{ν} (x)

(first kind) and

Y_{ν} (x)

(second kind) are the two linearly independent solutions of Bessel's differential equation, the canonical equation arising from the Laplacian in cylindrical coordinates. The key result:

J_{ν}

is finite at the origin and oscillates with slowly decaying amplitude, while

Y_{ν}

diverges to

- \infty

at the origin. They appear wherever waves, heat, or potentials live on circular or cylindrical domains. Introduced by Daniel Bernoulli and Euler, systematized by Friedrich Bessel in 1824.

Bessel functions are special functions defined as solutions to the second-order linear ODE $x^{2} y^{''} + x y^{'} + (x^{2} - ν^{2}) y = 0$ , where $ν$ is the order. They matter because separating the wave, heat, or Laplace equation in cylindrical or polar coordinates always produces this equation in the radial variable — making Bessel functions the cylindrical analogue of sines and cosines. The function of the first kind, $J_{ν} (x)$ , is regular at $x = 0$ ; the function of the second kind, $Y_{ν} (x)$ (also called the Neumann function $N_{ν}$ ), is singular there. A surprising fact: although they look like damped oscillations, their zeros are not periodic — the spacing only approaches $π$ asymptotically, encoding subtle information used in everything from drumhead acoustics to optical diffraction.

The Core Insight

Bessel's equation is "the radial wave equation." When you demand that a vibrating circular membrane or cylindrical field be a product $R (r) Θ (θ) Z (z) e^{iω t}$ , the angular and axial parts give sinusoids, but the radial part is forced to satisfy Bessel's equation. The two kinds, $J_{ν}$ and $Y_{ν}$ , are simply the "regular at center" and "singular at center" solutions — analogous to $cos$ and $sin$ , but adapted to the geometric fact that area grows like $r d r$ in the plane.

Formal Statement

Bessel's Differential Equation and the Two Kinds

Bessel's equation of order

ν

(with

ν \geq 0

real) is

x^{2} \frac{d ^{2} y}{d x ^{2}} + x \frac{d y}{d x} + (x^{2} - ν^{2}) y = 0.

⟳Separate the Helmholtz equation

\nabla^{2} u + k^{2} u = 0

in cylindrical coordinates; the radial ODE, after substituting

x = k r

, takes exactly this form.

Here $x$ is the (dimensionless) independent variable, $y (x)$ the unknown, and $ν$ the order governing the angular periodicity. The general solution is

y (x) = A J_{ν} (x) + B Y_{ν} (x),

⟳Equation [1] is a second-order linear ODE, so its solution space is two-dimensional;

J_{ν}

and

Y_{ν}

are a fundamental set since their Wronskian is nonzero.

where $A, B$ are constants fixed by boundary conditions.

The Bessel function of the first kind is defined by its power series

J_{ν} (x) = m = 0 \sum \infty \frac{( - 1 ) ^{m}}{m ! Γ ( m + ν + 1 )} (\frac{x}{2})^{2 m + ν},

⟳Substitute the Frobenius ansatz

y = \sum_{m} a_{m} x^{m + ν}

into [1], obtain the indicial equation

ν^{2} = ν^{2}

and the two-step recurrence

a_{m} = - a_{m - 2} / [m (m + 2 ν)]

, then normalize with the Gamma function.

The Bessel function of the second kind (Neumann function) is defined, for non-integer $ν$ , as

Y_{ν} (x) = \frac{J _{ν} ( x ) cos ( ν π ) - J _{- ν} ( x )}{sin ( ν π )},

⟳For non-integer

ν

J_{ν}

and

J_{- ν}

are independent; this particular combination is constructed so the limit

ν \to n

(integer) exists and yields a solution independent of

J_{n}

For integer order $n$ , equation [4] is taken as a limit $ν \to n$ , giving a logarithmic term.

Independence and the Wronskian

For all real $ν$ and $x > 0$ , the functions $J_{ν} (x)$ and $Y_{ν} (x)$ are linearly independent, with Wronskian

W {J_{ν}, Y_{ν}} (x) = J_{ν} (x) Y_{ν}^{'} (x) - J_{ν}^{'} (x) Y_{ν} (x) = \frac{2}{π x} .

⟳Write [1] in self-adjoint Sturm–Liouville form

(x y^{'})^{'} + (x - ν^{2} / x) y = 0

; Abel's identity gives

W = C / x

, and the constant

C = 2/ π

is fixed using the small-

x

behaviour of

J_{ν}

and

Y_{ν}

Since $W \neq = 0$ for $x > 0$ , equation [2] is the complete general solution.

Why It Holds / Derivation

Frobenius Construction of $J_\nu$

Step 1 — Indicial equation. Insert

y = \sum_{m \geq 0} a_{m} x^{m + s}

into [1]. The lowest-power term (

x^{s}

) yields

(s^{2} - ν^{2}) a_{0} = 0

, so the indicial roots are

s = + ν

and

s = - ν

. Step 2 — Recurrence. Matching the coefficient of

x^{m + s}

gives

[(m + s)^{2} - ν^{2}] a_{m} + a_{m - 2} = 0 ⟹ a_{m} = \frac{- a _{m - 2}}{m ( m + 2 s )} (s = ν) .

⟳

(m + ν)^{2} - ν^{2} = m (m + 2 ν)

; odd-indexed coefficients vanish since

a_{1} = 0

forced by the

m = 1

relation.

Step 3 — Solve the recurrence. Only even indices survive. Writing

m = 2 k

a_{2 k} = \frac{( - 1 ) ^{k} a _{0}}{2 ^{2 k} k ! ( ν + 1 ) ( ν + 2 ) \dots ( ν + k )} .

⟳Iterate [6]

k

times; the product in the denominator telescopes into a ratio of Gamma functions.

Step 4 — Normalize. Choose

a_{0} = 1/ [2^{ν} Γ (ν + 1)]

so the Gamma product becomes

Γ (k + ν + 1)

, producing series [3]. Step 5 — Second solution. Repeating with

s = - ν

gives

J_{- ν}

. For non-integer

ν

these are independent. For integer

n

, however,

J_{- n} (x) = (- 1)^{n} J_{n} (x)

, so a genuinely independent second solution must be built — this is

Y_{n}

, obtained from [4] by L'Hôpital's rule, which introduces a

ln x

singularity.

■

Friedrich Wilhelm Bessel | 1784–1846

A German astronomer and mathematician, Bessel was director of the Königsberg Observatory. While studying the perturbations of planetary orbits (specifically the problem of computing elliptical motion via Kepler's equation), he encountered and systematically analyzed the functions $J_{n}$ in his 1824 memoir Untersuchung des Theils der planetarischen Störungen. Though Daniel Bernoulli (vibrating chains, 1730s) and Euler (membranes, 1764) had met special cases earlier, Bessel gave the first comprehensive treatment of integer-order functions and their recurrence relations. He is equally famous for measuring the first stellar parallax (61 Cygni, 1838).

Building Intuition

Think of $cos x$ and $sin x$ as solutions of $y^{''} + y = 0$ — undamped oscillations. Bessel's equation [1] differs by the friction-like first-derivative term $x y^{'}$ . Dividing [1] by $x^{2}$ :

y^{''} + \frac{1}{x} y^{'} + (1 - \frac{ν ^{2}}{x ^{2}}) y = 0.

The

\frac{1}{x} y^{'}

term acts like a position-dependent damping that weakens as

x

grows, and the

ν^{2} / x^{2}

term is a centrifugal barrier near the origin. For large

x

, the damping fades and the equation looks like

y^{''} + y = 0

but with an amplitude that bleeds off as

1/ x

. Hence the asymptotic form

J_{ν} (x) \sim \frac{2}{π x} cos (x - \frac{ν π}{2} - \frac{π}{4}), x \to \infty.

⟳Substitute

y = u (x) / x

into the divided equation; the

1/ x

term cancels and you get

u^{''} + (1 - (ν^{2} - \frac{1}{4}) / x^{2}) u = 0

, which

\to u^{''} + u = 0

x \to \infty

So Bessel functions are "sinusoids on a stretched circular drumhead": they oscillate, but each successive ring of the drum carries less energy density, decaying like $1/ x$ .

$J_0$, $J_1$ (first kind) and $Y_0$, $Y_1$ (second kind). $J$'s are finite at $x=0$; $Y$'s plunge to $-\infty$.

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Worked Examples

Vibrating Circular Drumhead — Fundamental Frequency

Given: A circular drum of radius

a = 0.30

m, with the membrane fixed at the edge. Wave speed on the membrane

c = 100

m/s. Find the fundamental (lowest) vibration frequency. Setup. For the axisymmetric (radial,

ν = 0

) mode, displacement

\propto J_{0} (k r)

with

k = ω / c

. The fixed-edge condition

u (a) = 0

requires

J_{0} (k a) = 0.

Step 1 — First zero. The smallest positive root of

J_{0}

j_{0, 1} = 2.4048

. So

k a = 2.4048

. Step 2 — Wavenumber.

k = 2.4048/0.30 = 8.016 m^{- 1}

. Step 3 — Angular frequency.

ω = c k = 100 \times 8.016 = 801.6 rad/s

. Step 4 — Frequency.

f = ω / (2 π) = 801.6/6.2832 = 127.6 Hz

f_{0, 1} \approx 1.28 \times 1 0^{2} Hz

(Note: unlike a string, the overtones $j_{0, 2} = 5.520, j_{0, 3} = 8.654$ are not integer multiples — drums sound inharmonic.)

Series Evaluation of $J_0(1)$

Given: Compute

J_{0} (1)

from series [3] to 4 significant figures. Step 1 — Write terms (

ν = 0

, so

Γ (m + 1) = m!

J_{0} (1) = m = 0 \sum \infty \frac{( - 1 ) ^{m}}{( m ! ) ^{2}} (\frac{1}{2})^{2 m} .

Step 2 — Tabulate with

(1/2)^{2 m} = 4^{- m}

$m = 0$ : $1/1 = 1.000000$
$m = 1$ : $- \frac{1}{( 1 ) ^{2} \cdot 4} = - 0.250000$
$m = 2$ : $+ \frac{1}{( 2 ) ^{2} \cdot 16} = + 0.015625$
$m = 3$ : $- \frac{1}{( 6 ) ^{2} \cdot 64} = - 0.000434$
$m = 4$ : $+ \frac{1}{( 24 ) ^{2} \cdot 256} = + 0.0000068$

Step 3 — Sum:

1 - 0.25 + 0.015625 - 0.000434 + 0.0000068 = 0.765198

J_{0} (1) \approx 0.7652

(The tabulated value is $0.76519769$ — agreement confirms the rapid convergence of [3].)

Common Misconceptions

$J_{ν}$ and $Y_{ν}$ are NOT periodic. Their zeros only approach uniform spacing

π

x \to \infty

. Treating

J_{0}

as if its zeros were equally spaced (like

cos

) gives wrong overtone ratios for drums and wrong cutoff frequencies for waveguides.

Also note: $Y_{ν}$ must be discarded whenever the domain includes the origin (e.g., a solid disk), because $Y_{ν} (x) \to - \infty$ as $x \to 0^{+}$ . It is retained only for annular regions ( $r_{m i n} > 0$ ) where the origin is excluded.

One might guess that for integer order, $J_{- n}$ supplies a second independent solution (as $J_{- ν}$ does for non-integer $ν$ ). It does not. From series [3], one can show

J_{- n} (x) = (- 1)^{n} J_{n} (x),

J_{- n}

is just a scalar multiple of

J_{n}

— not independent. This degeneracy is precisely why

Y_{n}

(with its logarithmic singularity) had to be invented: it is the only way to complete the solution basis at integer order.

Connections

Bessel functions sit at a crossroads of analysis and physics. They are the cylindrical members of the family of Special Functions alongside Legendre and Hermite functions. They arise from Separation of Variables applied to the Helmholtz Equation and Laplace's Equation in cylindrical coordinates. Their orthogonality on $[0, a]$ with weight $r$ makes them a complete set for Fourier–Bessel Series, the cylindrical analogue of Fourier Series. Modified Bessel functions $I_{ν}, K_{ν}$ (solutions when the sign of the $x^{2}$ term flips) connect to Sturm–Liouville Theory and decay/growth problems. The integer-order $J_{n}$ are also the Fourier coefficients of $e^{i x s i n θ}$ , linking them to the Generating Functions formalism.

Real-World Applications

Acoustics of drums and cymbals — Modes of a circular membrane are $J_{n} (j_{n, m} r / a)$ ; the inharmonic zeros of $J_{n}$ explain why drums lack a clear pitch.
Electromagnetic waveguides — TE/TM modes in a circular cylindrical waveguide have transverse fields $\propto J_{n}$ ; cutoff frequencies are set by zeros of $J_{n}$ (TM) or $J_{n}^{'}$ (TE), critical in microwave engineering.
Optical diffraction (Airy pattern) — Light through a circular aperture produces an intensity $\propto [2 J_{1} (x) / x]^{2}$ , fixing the diffraction-limited resolution of telescopes and microscopes (the Rayleigh criterion uses $j_{1, 1} = 3.8317$ ).
Heat conduction in cylinders — Transient temperature in a cylindrical rod or wire solves the heat equation via Fourier–Bessel series in $J_{0}$ .
FM radio and signal processing — Frequency-modulated signals expand as $\sum_{n} J_{n} (β) cos (\dots)$ ; Bessel coefficients give the sideband amplitudes (Carson's rule uses where $J_{n} (β)$ becomes negligible).
Quantum mechanics — The free-particle radial wavefunction in 2D cylindrical or 3D spherical (via spherical Bessel functions $j_{ℓ}$ ) symmetry uses Bessel functions; also the infinite cylindrical well.
Vibration of suspended cables and chains — Daniel Bernoulli's original 1730s problem of a hanging chain's normal modes is solved by $J_{0}$ .

Historical Development

1732–1734 — Daniel Bernoulli analyzes the oscillations of a heavy hanging chain, implicitly encountering $J_{0}$ .
1764 — Leonhard Euler studies the vibrations of a stretched circular membrane and writes down series for $J_{n}$ of arbitrary integer order.
1770 — Joseph-Louis Lagrange uses Bessel-type series in his work on planetary orbital perturbations.
1816 — Bessel begins systematic analysis of the functions in connection with Kepler's equation.
1824 — Friedrich Bessel publishes his foundational memoir, giving recurrence relations and integral representations for integer-order $J_{n}$ ; the functions take his name.
1867 — Carl Neumann introduces the function of the second kind $Y_{n}$ (hence "Neumann function," symbol $N_{n}$ ), supplying the missing second solution.
1868 — Hermann Hankel develops the functions $H_{ν}^{(1, 2)} = J_{ν} \pm i Y_{ν}$ for radiating/incoming cylindrical waves.
1871–1874 — Eugen Lommel writes the first comprehensive treatise, Studien über die Bessel'schen Functionen, unifying the theory.
1922 — G. N. Watson publishes A Treatise on the Theory of Bessel Functions, still the definitive reference.

Subtleties & Limits

The distribution of Bessel zeros remains a source of deep results and open refinements. While McMahon's asymptotic expansion gives the

m

-th zero of

J_{ν}

j_{ν, m} \approx β - \frac{4 ν ^{2} - 1}{8 β}