Uniform enclosures for the phase and zeros of Bessel functions and their derivatives

Abstract

We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.

Figures (13)

• Figure 1: Plots of $\mathcal{C}_{\nu, \tau}(x)$.
• Figure 2: Plots of $\theta_\nu(x)$. The filled colour-coded dots on the horizontal axis indicate the positions of zeros $j_{\nu,k}$ and the hollow dots the positions of zeros $y_{\nu,k}$. The phase functions are calculated using the method of Hor.
• Figure 3: Plot of $\tau^*_\nu$ against $\nu$.
• Figure 4: Plots of $\mathcal{C}'_{\nu, \tau}(x)$.
• Figure 5: Plots of $\varphi_\nu(x)$. The filled colour-coded dots on the horizontal axis indicate the positions of zeros $j'_{\nu,k}$ and the hollow dots the positions of zeros $y'_{\nu,k}$. The phase functions are calculated using the method of Hor.

Theorems & Definitions (33)

• Remark 1.1
• Example 1.2
• Remark 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• Corollary 1.7
• Lemma 1.8
• Remark 1.9
• Theorem 1.10