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ISEGORIABenjamin Haire
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Complex analysis · special functions

Two series.
One hidden geometry.

Laurent series lets a function change its vocabulary when you cross a singularity. Bessel functions turn that same discipline of local expansion into waves, zeros, and resonance.

The theory

A compact dictionary for both series.

Keep the centre, the singularities, and the convergence region visible. Those three facts tell you which expansion is legal.

Laurent

Coefficients come from a circle.

Around a centre z₀, a function may have both positive and negative powers. The coefficient is a contour average, so the annulus is part of the definition.

\[f(z)=\sum_{n=-\infty}^{\infty}a_n(z-z_0)^n\]\(a_n=\frac{1}{2\pi i}\oint_\gamma\frac{f(\zeta)}{(\zeta-z_0)^{n+1}}d\zeta\)
Convergence

The nearest singularity sets the radius.

A Taylor series fills a disk. A Laurent series fills the gap between two circles: the inner and outer radii are distances to singularities.

\[R_{\mathrm{inner}}<|z-z_0|<R_{\mathrm{outer}}\]outside the poles, the same function can need a new vocabulary
Bessel

A radial ODE chooses the series.

Bessel's equation appears when separation of variables meets circular geometry. Regularity at x = 0 selects Jν; a second independent solution carries the singular behaviour.

\[x^2y''+xy'+(x^2-\nu^2)y=0\]\[J_\nu(x)=\sum_{m=0}^{\infty}\frac{(-1)^m}{m!\,\Gamma(m+\nu+1)}\left(\frac{x}{2}\right)^{2m+\nu}\]
Worked examples

Follow the algebra, then touch the plot.

These examples use the same functions as the controls below. The highlighted step is the one the visual makes easier to see.

Example A · Laurent

Expand \(f(z)=\frac{1}{z(z-1)}\) near 0

1
Rewrite the denominator: \(f(z)=-\frac{1}{z}\frac{1}{1-z}\).
2
Use the geometric series when \(|z|<1\): \(\frac{1}{1-z}=\sum_{n=0}^{\infty}z^n\).
3
\[f(z)=-z^{-1}-1-z-z^2-\cdots\]For the outer annulus, rewrite instead in powers of \(z^{-1}\). Move the radius slider across 1 to see the switch.
Example B · Bessel

Approximate \(J_0(2)\) with four terms

1
Set \(\nu=0\), \(x=2\): each term becomes \( (-1)^m/(m!)^2\).
2
\[J_0(2)\approx1-1+\frac{1}{4}-\frac{1}{36}=0.2222\ldots\]
3
The next terms pull the answer toward \(0.2238908\ldots\). Increase “series terms” and watch the zeros settle as the tail shrinks.
01 / Laurent series

An annulus chooses the powers.

The same rational function has two honest expansions. Move the radius across its pole at z = 1 and the vocabulary flips from positive to negative powers.

Laurent series on the complex planeDomain colouring of 1/[z(z−1)] with its poles at 0 and 1, the boundary circle |z| = 1, the inspection circle and partial-sum samples coloured by error.
  • colour = arg f(z), bands = |f| doubling
  • partial sum agrees
  • partial sum is off
  • |z| = 1, between the annuli

Each point of the plane is coloured by the phase of f(z), with the brightness bands marking every doubling of |f|; the colours wind once around each simple pole. The faded region is where the chosen expansion does not converge. The strip underneath follows the error of the partial sum around your circle: it peaks at θ = 0, the point nearest the pole at z = 1.

Inspect the annulus

For f(z) = 1 / [z(z − 1)], the function of example A, choose a radius and watch the valid series change.

0worst error on the circle
8terms in the partial sum
02 / Bessel functions

A power series that becomes a wave.

Jν(x) is assembled from alternating even steps. The factorial and gamma terms tame the tail; the sign changes create oscillation.

Bessel function graphThe truncated series for J nu, the true function, its decaying envelope, and the zeros of both.
  • truncated series
  • true Jν
  • envelope ±√(2/πx)
  • zeros of the series
  • true zeros

A finite sum of powers must eventually run off to infinity, so every truncation fails somewhere. The shaded band shows where this one has: from there on its zeros are artefacts. The violet envelope is the large-x behaviour, a cosine whose amplitude falls like 1/√x. Hover the plot to read both curves.

Tune the wave

Change the order ν and the number of retained terms. The graph and the term ledger share one model.

Jν(x) = Σ (−1)m (x/2)2m+ν / [m! Γ(m+ν+1)]m = 0, 1, 2, … · entire in x for integer order; noninteger order needs a branch
term ledger at x = 4
Convergence and truncation
03 / Drum modes

Bessel zeros you can watch vibrate.

Separate the wave equation on a circular membrane and the radial part is Jn. The rim must stay still, so the allowed shapes stretch Jn until one of its zeros lands exactly on the edge.

Choose a mode

u(r, θ, t) = Jn(jn,m r) cos(nθ) cos(ωt), with jn,m the m-th zero of Jn and ω proportional to jn,m.

angular order n (nodal diameters)
radial index m (which zero)
jn,m
frequency ÷ fundamental
Nodes
The bridge

Local coordinates matter.

A Taylor series forbids negative powers because it must be regular at its centre. A Laurent series allows them because a punctured neighbourhood has a hole.

−1
residue of 1 / [z(z − 1)] at z = 0
The check

Count the oscillations.

For large x, Bessel waves behave like a cosine with a slowly shrinking envelope. The zeros are the geometry of that interference.

6
visible zero crossings at ν = 0