On the computation of zeroes of 𝐽_{𝑛}(𝑧)-𝑖𝐽_{𝑛+1}(𝑧)=0

MacDonald, D. A.

doi:10.1090/qam/1486539

On the computation of zeroes of $J_n(z)-iJ_{n+1}(z)=0$

Author: D. A. MacDonald
Journal: Quart. Appl. Math. 55 (1997), 623-633
MSC: Primary 33C10; Secondary 65H10
DOI: https://doi.org/10.1090/qam/1486539
MathSciNet review: MR1486539
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Abstract | References | Similar Articles | Additional Information

Abstract: The roots of the equation

$\displaystyle J_n^2(z) + J_{n + 1}^2(z) = 0$

, in which

is a positive integer or zero, are of interest to the specialist in wave reflection from multi-sloped beaches [1]. This note shows how to obtain accurate roots of the equation when

is not large.

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[2] G. N. Watson, Theory of Bessel Functions, Cambridge University Press, 1944 MR 0010746
[3] D. A. MacDonald, The roots of ${J_0}\left( z \right) - i{J_1}\left( z \right)$ , Quart. Appl. Math. XLVII, 375-378 (1989) MR 998110
[4] M. Renardy, Problems and Solutions, Ed. M. Klamkin, Siam Review, vol. 31, 1989, pp. 126-127