The roots of $J_0(z)-iJ_1(z)=0$
Author:
D. A. Macdonald
Journal:
Quart. Appl. Math. 47 (1989), 375-378
MSC:
Primary 33A40
DOI:
https://doi.org/10.1090/qam/998110
MathSciNet review:
998110
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Abstract: Synolakis [1] has proved that the equation \[ {J_0}\left ( z \right ) - i{J_1}\left ( z \right ) = 0\] has no zeros in the half plane Im $z > 0$. In this note a table of the first thirty roots, correct to $O\left ( {{{10}^{ - 6}}} \right )$, is presented and an asymptotic formula, which is correct to better than one tenth of one percent for the smallest zero, is derived.
- Costas Emmanuel Synolakis, On the roots of $f(z)=J_0(z)-iJ_1(z)$, Quart. Appl. Math. 46 (1988), no. 1, 105–107. MR 934685, DOI https://doi.org/10.1090/S0033-569X-1988-0934685-0
- George F. Carrier, Gravity waves on water of variable depth, J. Fluid Mech. 24 (1966), 641–659. MR 200009, DOI https://doi.org/10.1017/S0022112066000892
C. E. Synolakis, The runup of solitary waves, J. Fluid Mech. 185, 523–545 (1987)
- E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469
C. E. Synolakis, On the roots of $f\left ( z \right ) = {J_0}\left ( z \right ) - i{J_1}\left ( z \right )$, Quart. Appl. Math. 46, 105–107 (1988)
G. F. Carrier, Gravity waves of water of variable depth, J. Fluid Mech. 24, 641–659 (1966)
C. E. Synolakis, The runup of solitary waves, J. Fluid Mech. 185, 523–545 (1987)
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, 1927
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Article copyright:
© Copyright 1989
American Mathematical Society
