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On the complex zeros of $H\sb \mu(z)$, $J\sb \mu'(z)$, $J\sb \mu''(z)$ for real or complex order. (English) Zbl 0756.33001
The authors establish some propositions about the nonexistence of complex zeros of the functions $H\sb \mu(z)$, $J\sb \mu'(z)$ and $J\sb \mu''(z)$, for $\mu$ in general complex. Some bounds for the purely imaginary zeros of the above functions are also obtained assuming their existence. These bounds for the purely imaginary zeros of $H\sb \mu(z)$ are as follows: $$\align \rho\sb 2 & >-2(\mu\sb 1+1)\vert\alpha+\mu\vert\sp 2/(\mu\sb 1+\alpha\sb 1), \qquad -1<\mu\sb 1<-\alpha\sb 1,\\ \vert\rho\sb 2\vert & >(\vert\mu+\alpha\vert \sqrt{\vert\mu+\alpha\vert\sp 2+2\mu\sb 2(\mu\sb 2+\alpha\sb 2)}-\vert\mu+\alpha\vert\sp 2) /\vert\mu\sb 2+\alpha\sb 2\vert\qquad \text{and}\\ \mu\sb 2 & >\max\{0,-\alpha\sb 2\}\qquad \text{or}\qquad \mu\sb 2<\min\{0,-\alpha\sb 2\},\endalign$$ where $\rho\sb 1$, $\rho\sb 2$, $\mu\sb 1$, $\mu\sb 2$, $\alpha\sb 1$, $\alpha\sb 2$ are the real and imaginary parts of $\rho$, $\mu$ and $\alpha$. The authors use these bounds to measure the purely imaginary zeros of $J\sb \mu''(z)$. The results proved by the authors generalize some of the results given earlier by {\it E. K. Ifantis}, {\it P. D. Siafarikas} and {\it C. B. Kouris} [J. Math. Anal. Appl. 104, 454-466 (1984; Zbl 0558.34006)].

33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
Full Text: DOI
[1] Ifantis, E. K.; Siafarikas, P. D.: Ordering relations between the zeros of miscellaneous Bessel functions. Appl. anal. 23, 85-110 (1986) · Zbl 0597.33009
[2] Ifantis, E. K.; Siafarikas, P. D.; Kouris, C. B.: Conditions for solution of a linear first-order differential equation in the Hardy-Lebesgue space and applications. J. math. Anal. appl. 104, 454-466 (1984) · Zbl 0558.34006
[3] Ifantis, E. K.; Siafarikas, P. D.; Kouris, C. B.: The imaginary zeros of a mixed Bessel function. J. appl. Math. phys. 39, 157-165 (1988) · Zbl 0651.33006
[4] Watson, G. N.: A treatise on the theory of Bessel functions. (1958) · Zbl 0083.20702