Bessel Function - Bourget's Hypothesis

Bourget's Hypothesis

Bessel himself originally proved that for non-negative integers n, the equation J_n(x) = 0 has an infinite number of solutions in x. When the functions J_n(x) are plotted on the same graph, though, none of the zeros seem to coincide for different values of n except for the zero at x = 0. This phenomenon is known as Bourget's hypothesis after the nineteenth century French mathematician who studied Bessel functions. Specifically it states that for any integers n ≥ 0 and m ≥ 1, the functions J_n(x) and J_n+m(x) have no common zeros other than the one at x = 0. The hypothesis was proved by Carl Ludwig Siegel in 1929.