Concept

Bessel function

A solution of Bessel's equation, the functions Jₙ that describe waves and vibrations with circular symmetry. They arise as averages of e^(i(x sin t − nt)) over a turn, so they appear wherever an oscillating phase is integrated round a circle or over a ball.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Rebuilding a ball at its own centre. Fourier inversion of the indicator of the unit segment, disc and ball, keeping frequencies below R, evaluated at the centre: at R = 8 the values are 0.9873, 0.9206, 0.9873; in three dimensions at R = 7.75 it is 1.6369.

A ball rebuilt everywhere but its centre

Rebuild a solid ball from its frequencies, keeping those below a cut-off, and read the result at the centre, where the ball is perfectly smooth. In the plane the answer settles on 1. In space it swings between 0.36 and 1.64 for ever, because the whole surface of the ball is the same distance from the centre and its jump arrives there in step.

analysis · Fourier series

The curve whose bending grows with its length. Euler's spiral, curvature equal to arc length, for s from −7 to 7, winding into ±(0.88623, 0.88623).

The curve that bends as far as it has gone

Say how sharply a curve bends at each distance along it, and the curve is fixed. Bending in proportion to distance gives Euler's spiral, whose arms are infinitely long and end a finite distance away. Bending that repeats decides at once whether the curve ever closes, and taken in steps the spiral is a Gauss sum.

analysis · Arc length

Named alongside it

The objects these essays reach for when they reach for this one.

Arc length Curvature Differential equation Dimension Dirichlet kernel Fourier series Fourier transform Fresnel integral Gauss sum Gibbs' phenomenon Pointwise convergence Spiral

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