Summary
LetC vk be thekth positive zero of the cylinder functionC v(x)=cosαJ v(x)−sinαY v(x), whereJ v(x),Y v(x) are the Bessel functions of first kind and second kind, resp., andv>0, 0≦α<π. Definej vk byj vk=C vk with\(\kappa = k - \frac{\alpha }{\pi }\). Using the notation 1/K=ε, we derive the first two terms of the asymptotic expansion ofj vk in terms of the powers of ε at the expense of solving a transcendental equation. Numerical examples are given to show the accuracy of this approximation.
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References
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Dedicated to the memory of Professor Lothar Collatz
This work has been supported by the Hungarian Scientific Grant No. 6032/6319
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Elbert, Á. An approximation for the zeros of Bessel functions. Numer. Math. 59, 647–657 (1991). https://doi.org/10.1007/BF01385801
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DOI: https://doi.org/10.1007/BF01385801