The roots of 𝐽₀(𝑧)-𝑖𝐽₁(𝑧)=0

Abstract

Synolakis [1] has proved that the equation \[ J 0 ( z ) − i J 1 ( z ) = 0 {J_0}\left ( z \right ) - i{J_1}\left ( z \right ) = 0 \] has no zeros in the half plane Im z > 0 z > 0 . In this note a table of the first thirty roots, correct to O ( 10 − 6 ) O\left ( {{{10}^{ - 6}}} \right ) , is presented and an asymptotic formula, which is correct to better than one tenth of one percent for the smallest zero, is derived.