Electrostatics and zeros of Bessel functions

We discuss an electrostatic interpretation of the zeros of the Bessel function Jv(z) where v is an unrestricted real variable. It is known that the zeros are real when v⩾−1 and that more and more complex zeros appear when v decreases through values less than −1. The dynamic behaviour of the zeros as time t = −v increases can be modelled by the behaviour of a suitably normalized infinite set of particles of charge +1 in the presence of a varying charge v + 12 at the origin. The idea for this model comes from the work of Stieltjes, who interpreted the zeros of Jacobi and other orthogonal polynomials as equilibrium positions in certain one-dimensional electrostatic problems. Similar interpretations can be given for the way in which the zeros of J′v(z) vary with v.