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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 &gt; 0 z &gt; 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.

The roots of $J_0(z)-iJ_1(z)=0$