Abstract
In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ vk of the derivative of the general Bessel functionC v(x)=J v(x)cosα−Y v(x)sinα, 0≤α<π, whereJ v(x) andY v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ vk>∥v∥,c′ vk increases asv increases. Here we prove several additional properties forc′ vk. Our main result is thatc′ vk is concave as a function ofv, whenc′ vk>∥v∥>0. This implies the concavity ofc′ vk for everyk=2,3, ⋯. In the case of the zerosJ′ vk of dxd J v(x) we extend this property tok=1 for everyv≥0.
Sommario
In questo lavoro il nostro interesse è rivolto al comportamento, rispetto av, delk-esimo zeroc′ vk della derivata della funzione cilindricac′ vk(x)=J v(x)cosα−Y v (x)sinα, 0≤α<π, doveJ v(x) eY v(x) indicano le funzioni di Bessel rispettivamente di prima e seconda specie. E' ben noto che nel casoc′ vk>∥v∥,c′ vk è una funzione crescente div.
Qui, proviamo parecchie ulteriori proprietà per la funzionec′ vk. Il principale risultato è chec′ vk è concava rispetto av, perc′ vk>∥v∥>0. Questo implica la concavità dic′ vk per ognik=2,3,⋯. Nel caso degli zerij′ vk della funzione dkd J v(x) possiamo estendere questo proprietà anche ak=1, per ogniv>0.
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Work sponsored by the Consiglio Nazionale delle Ricerche — Italy.
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Elbert, Á., Laforgia, A. On the zeros of derivatives of Bessel functions. Z. angew. Math. Phys. 34, 774–786 (1983). https://doi.org/10.1007/BF00949055
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DOI: https://doi.org/10.1007/BF00949055