Geodesic
On zeros of the combination of products of Bessel functions
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 60 (2019), pp. 5-10 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper, the function $f_\nu(t)=J_\nu(t)I_{-\nu}(t)+I_\nu(t)J_{-\nu}(t)$, $0<\nu<1$, $\mathrm{Re}\,t>0$, is investigated. Such functions were little studied in the literature. It is proved that more general functions $f_{\nu,\mu}^{(1),(2)}(t)=J_\nu(t)I_{-\mu}(t)\pm I_\mu(t)J_{-\nu}(t)$ have a countable set of real zeros and a countable set of pure imaginary zeros. The proof uses the well-known Sturm theorem for second-order differential equations. The statement is applied to specific examples. In the case $\nu=1/2$, the function $f_{1/2}(x)=J_{1/2}(x)I_{-1/2}(x)+I_{1/2}(x)J_{-1/2}(x)$ is reduced to an elementary function $f_{1/2}(x)=\frac2{\pi x}(\sin x\cdot\cosh x+\cos x\cdot\sinh x)$, and an asymptotic formula for its positive zeros $x=-\frac\pi4+\pi k+O(e^{-2\pi k})$ is found. Function $\hat{f}_{1/2}(x)=J_{1/2}(x)I_{-1/2}(x)-I_{1/2}(x)J_{-1/2}(x)$ has the following positive zeros: $x=\frac\pi4+\pi k+O(e^{-2\pi k})$.
Keywords: Bessel function, modified Bessel function, set of zeros of the function, Sturm theorem. 
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A. A. Gimaltdinova; E. P. Anosova. On zeros of the combination of products of Bessel functions. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 60 (2019), pp. 5-10. http://geodesic.mathdoc.fr/item/VTGU_2019_60_a0/

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